CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY
EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 33 DK2800 Lyngby, Denmark e-mail: [email protected] September 1999 KRZYSZTOF CIESIELSKI Mathematics Institute Jagiellonian University Reymonta 4 EMS News : Committee and Agenda ...... 2 30-059 Krakow, Poland e-mail: [email protected] Editorial by EMS Vice-President Andrzej Pelczar ...... 3 KATHLEEN QUINN Open University [address as above] Fourth Diderot Mathematical Forum : Mathematics and Music ...... 4 e-mail: [email protected] SPECIALIST EDITORS Introducing the Editorial Team : Part 1 ...... 5 INTERVIEWS EMS Council Meeting in Barcelona: First Announcement ...... 6 Steen Markvorsen [address as above] SOCIETIES Oxford Doctorate for Andrew Wiles ...... 7 Krzysztof Ciesielski [address as above] EDUCATION Interview with Tim Gowers ...... 8 Vinicio Villani Dipartimento di Matematica ICIAM99 in Edinburgh ...... 10 Via Bounarotti, 2 56127 Pisa, Italy SIAM and EMS Joint Conference on Computational Science ...... 11 e-mail: [email protected] A Universal Mathematical Resources Locator? ...... 12 MATHEMATICAL PROBLEMS Paul Jainta 1999 Anniversaries : Caspar Wessel ...... 13 Werkvolkstr. 10 D-91126 Schwabach, Germany 1999 Anniversaries : E. C. Titchmarsh ...... 16 e-mail: [email protected] ANNIVERSARIES Societies Corner : Swiss Mathematical Society ...... 18 June Barrow-Green and Jeremy Gray Open University [address as above] Societies Corner : Edinburgh Mathematical Society ...... 20 e-mail: [email protected] and [email protected] and Forthcoming Conferences ...... 22 CONFERENCES Recent Books ...... 26 Kathleen Quinn [address as above] RECENT BOOKS Designed and printed by Armstrong Press Limited Ivan Netuka and Vladimir Souèek Mathematical Institute Unit 3 Crosshouse Road, Southampton, Hampshire SO14 5GZ, UK Charles University phone: (+44)-23-8033 3132; fax: (+44)-23-8033 3134 Sokolovská 83 Published by European Mathematical Society 18600 Prague, Czech Republic ISSN 1027 - 488X e-mail: [email protected] and [email protected] NOTICE FOR MATHEMATICAL SOCIETIES ADVERTISING OFFICER Labels for the next issue will be prepared during the second half of August. Martin Speller Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department of Mathematics, Department of Mathematics P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail: [email protected] Glasgow Caledonian University Glasgow G4 0BA, Scotland INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTER e-mail: [email protected] Institutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat, OPEN UNIVERSITY PRODUCTION Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e-mail: TEAM Please include the name and full address (with postal code), telephone and fax number (with coun- try code) and e-mail address. The annual subscription fee (including mailing) is 60 euros; an Kathleen Quinn, Liz Scarna invoice will be sent with a sample copy of the Newsletter.
EMS June 1999 1 EMS NEWS EMS News: Committee and Agenda EXECUTIVE COMMITTEE PRESIDENT (1999–2002) Prof. ROLF JELTSCH EMS Agenda Seminar for Applied Mathematics 1999 ETH, CH-8092 Zürich, Switzerland e-mail: [email protected] 30 September VICE-PRESIDENTS Deadline for submission of proposals for the 2001 EMS Summer Schools Prof. ANDRZEJ PELCZAR (1997–2000) contact: David Brannan, e-mail: [email protected] Institute of Mathematics 9 - 10 October Jagellonian University Executive Committee Meeting, hosted by the Swiss Mathematical Society and ETH, Raymonta 4 PL-30-059 Krakow, Poland Zurich (Switzerland) e-mail: [email protected] 15 November Prof. LUC LEMAIRE (1999–2002) Deadline for submission of material for the December issue of the EMS Newsletter Department of Mathematics contact: Robin Wilson, e-mail: [email protected] Université Libre de Bruxelles C.P. 218 – Campus Plaine December Bld du Triomphe Second announcement of the Third European Congress of Mathematics (3ecm), Barcelona B-1050 Bruxelles, Belgium (Spain) e-mail: [email protected] contact: S. Xambó-Descamps, e-mail: [email protected] SECRETARY (1999–2002) website: www.iec.es/3ecm/ Prof. DAVID BRANNAN Department of Pure Mathematics 3 - 4 December The Open University Fourth Diderot Mathematical Forum, on Mathematics and Music, in Lisbon (Portugal), Walton Hall Paris (France) and Vienna (Austria) Milton Keynes MK7 6AA, UK contact: Mireille Chaleyat-Maurel, e-mail: [email protected] e-mail: [email protected] TREASURER (1999–2002) Prof. OLLI MARTIO 2000 Department of Mathematics 31 January P.O. Box 4 Nominations to the Secretariat for delegates of individual members (details to appear in FIN-00014 University of Helsinki the December Newsletter) Finland contact: EMS Secretariat, e-mail: [email protected] e-mail: [email protected] ORDINARY MEMBERS February - March Prof. BODIL BRANNER (1997–2000) Voting for delegates of individual members Mathematical Institute contact: EMS Secretariat, e-mail: [email protected] Technical University of Denmark Building 303 15 February DK-2800 Lyngby, Denmark Deadline for submission of material for the March issue of the EMS Newsletter e-mail: [email protected] contact: Robin Wilson, e-mail: [email protected] Prof. DOINA CIORANESCU (1999-2002) 25 - 26 March Laboratoire d’Analyse Numérique Executive Committee Meeting, hosted by the Polish Mathematical Society and the Université Paris VI Institute of Mathematics of the Polish Academy of Sciences, Bedlevo, near Pozñan 4 Place Jussieu (Poland) 75252 Paris Cedex 05, France e-mail: [email protected] 15 May Prof. RENZO PICCININI (1999–2002) Deadline for submission of material for the June issue of the EMS Newsletter Dipto di Matem. F. Enriques contact: Robin Wilson, e-mail: [email protected] Universit à di Milano Via C. Saldini 50 3 – 7 July I-20133 Milano, Italy ALHAMBRA 2000: a joint mathematical European-Arabic conference in Granada e-mail: [email protected] (Spain), promoted by the European Mathematical Society and the Spanish Royal Prof. MARTA SANZ-SOLÉ (1997–2000) Mathematical Society Facultat de Matematiques contact: Ceferino Ruiz, e-mail: [email protected] Universitat de Barcelona website: www.ugr.es/~ruiz/ Gran Via 585 E-08007 Barcelona, Spain 7 – 8 July e-mail: [email protected] Council Meeting in Barcelona (Spain) Prof. ANATOLY VERSHIK (1997–2000) contact: EMS Secretariat, e-mail: [email protected] P.O.M.I., Fontanka 27 10 – 14 July 191011 St Petersburg, Russia Third European Congress of Mathematics (3ecm) in Barcelona (Spain) e-mail: [email protected] contact: S. Xambó-Descamps, e-mail: [email protected] EMS SECRETARIAT website: www.iec.es/3ecm/ Ms. T. MÄKELÄINEN Department of Mathematics 24 July – 3 August P.O. Box 4 EMS Summer School in Edinburgh (Scotland): New analytic and geometric methods in inverse FIN-00014 University of Helsinki problems Finland organiser and contact: Erkki Somersalo (Otaniemo, Finland), e-mail: [email protected] tel: (+358)-9-1912-2883 fax: (+358)-9-1912-3213 17 August – 2 September telex: 124690 EMS Summer School at Saint-Flour, Cantal (France): Probability theory e-mail: [email protected] organiser and contact: Pierre Bernard (Clermont-Ferrand, France), e-mail: bernard@ website: http://www.emis.de ucfma.univ.bpclermont.fr 2 EMS September 1999 EDITORIAL EMS statutes should be rediscussed, interest in mathematical education in the revised and then ‘stabilised’, in some sense, ‘European dimension’, especially with as soon as the number of individual mem- respect to a general reform of educational bers became large enough; for details, see systems (including fundamental reforms in EditorialEditorial the EMS statutes. Since the growth in the mathematics teaching) in several Central individual membership has been slower and Eastern European countries, in order than expected, this decision is still to be to be aware of what has already been done by implemented; one can say that at that time and what we can expect in that area in the the process of building up and shaping the near future. EMS Society will be barely completed. Such an Returning to the question of member- ship, let me add a historical remark about the Polish Mathematical Society. During Vice-President its foundation meeting in Kraków in April 1919 (see the Societies section in the June Andrzej Pelczar issue), two founder members, Stanis³aw Zaremba and Stefan Banach, decided after (Kraków) discussion that the Society should have a purely scientific character; all suggestions In October 2000 the European proposing the popularisation of mathe- Mathematical Society celebrates the tenth matics and other not strictly scientific activ- anniversary of its creation. Looking back ities were officially rejected. Eventually, on the past nine years we can try to evalu- after some decades, the Society changed its ate the achievements of the EMS, counting identity and became an association that its successes, pointing out possible mis- included in its mission the popularisation takes or unrealised plans while concentrat- of mathematics and (later) the teaching of ing on projects for the future. Such an mathematics. The membership rules now anniversary provides a good occasion for a allow schoolteachers to be members of the discussion on some problems concerning Polish Mathematical Society. the general shape of the Society and of its The above remarks are of minor impor- activities, as for instance its role in the tance with respect to membership of the preparation of European Congresses of EMS, but they raise a more important Mathematics and the Diderot problem, that of understanding the word Mathematical Forums, the patronage mathematician. This seems to be important offered to scientific conferences, and so on. for numerous reasons; let me mention only Photo by Stefan Ciechan one – statistical data indicating the num- Important aspects of the EMS’s activity, published with kind permission of Forum Akademickie and more generally of the presence of the bers of mathematicians in European coun- Society on the European scientific and cul- opinion is not based on an official inter- tries. In Poland we have two so-called ‘pro- tural platform, concern the use of pretation of the statutes and need not be fessional titles’, licencjat and magister, equiv- European financial sources; compare, for shared by everyone; on the other hand, it alent to a B.Sc. and an M.Sc. in English or instance, Luc Lemaire’s Editorial in the is not totally unacceptable, and is formu- American terminology. Licencjat is purely June issue (EMS Newsletter 32). lated here as ‘intellectual provocation’ in a ‘professional’ title, while magister is the It would be practically impossible and order to point out that the most important first step in an academic career: everyone superfluous – in view of Rolf Jeltsch’s question concerning individual members wishing to get a Ph.D. must have the ‘title’ Editorial in the March issue (EMS is: how do we increase their number? magister. In official (administrative) termi- Newsletter 31) – to present here all the On the quantitative development of the nology everybody who gets the profession- forms of EMS activity related to research in EMS we have to think about the possible al title of licencjat in mathematics (or, of mathematics and its applications (with all impact of the growth in the individual course, the higher one: magister) is counted its interdisciplinary aspects), international membership on the future shape of the in the statistical data as a mathematician. cooperation of mathematicians and Society. The EMS can be briefly charac- In all databases concerning higher educa- national societies (including, but not terised as a scientific society that is also tion only mathematicians with an M.Sc. are restricted to, support for young mathe- concerned with the popularisation of counted as members of staff of a higher maticians from countries facing financial mathematics as well as its applications and education institution. It is probable that difficulties) and everything that has been (at least implicitly) with the teaching of future databases for higher education will done, according to the general mission of mathematics, stressing to the public the indicate only those members of academic the Society, for the popularisation of math- role played by mathematics in science, cul- staff with a Ph.D. Thus, in the future, from ematics, with special attention to the role ture, technology and civilisation in gener- a ‘statistical point of view’, mathematicians of mathematics in the evolution of culture, al. It is natural that these tasks and goals in higher education institutions in Poland technology and civilisation. It would also were first realised by researchers and will all be doctors in mathematics. (I think be useless to try to describe all the present teachers in higher education institutions; that the terminology varies from country to and future problems of the Society and the at present the individual members come country.) EMS’s role in the scientific and cultural life almost exclusively from these categories. It would be very useful to obtain more of Europe; such a task would be too ambi- However, since the membership of several precise terminology on a European scale. tious and beyond the author’s competence. national mathematical societies (corporate It would be valuable in particular to con- I will thus limit myself to a few of them, members of the EMS) also includes school- tinue the extremely interesting panel dis- selected in a quite arbitrary and subjective teachers and other mathematical ‘alumni’ cussion on ‘Demography of mathemati- way. not working as scientists or academics, we cians’ at the Second European Congress of The first group of problems concerns must expect that some of them will become Mathematics (Budapest, 1996). Maybe the the members of the EMS. Everyone knows future members of the EMS – we should EMS should suggest something along that the membership is very heteroge- welcome them warmly. In the long run those lines to stimulate some clarification. neous with its three categories of members. this should change the future image of the The first step would be to compare the ter- National societies, as corporate founder Society. I think that the main impact minology in various countries. members, created the Society in M¹ dralin would be a stronger interest by the EMS in The next question I regard as important (Poland) in October 1990. Expecting a the teaching of mathematics (at all levels, since it touches on one form of activity that rapid growth in the number of individual including elementary and secondary). makes the Society visible. Among several members, these societies decided that the Even now the Society should extend its interesting problems discussed by mem- EMS June 1999 3 EDITORIAL bers of the EMS Council at the Budapest Congresses present prizes to young mathe- respect. We add that the acceptance of the meeting, one concerned a fundamental maticians, and there are round table dis- idea of the European Congresses has been question: is the organisation of large and cussions on special topics. We note finally emphasised in spectacular form by the ‘non-specialised’ conferences, such as that progress in mathematics, and the strong competition for permission to International Congresses of growth in numbers of researchers and organise the Third European Congress of Mathematicians or the European mathematical scientific centres – ‘density Mathematics; Barcelona was the winner of Congresses of Mathematics, really reason- in the time scale’ of mathematically impor- that competition. I am sure that the Third able and fruitful? A ‘weaker’ version of this tant events – make it reasonable to organ- Congress will be fully successful and will problem questioned the need to organise ise large and prestigious congresses inde- dismiss all doubts concerning the organisa- European Congresses in view of the fact pendently of the ICMs. European tion of European Congresses. that an ICM is organised every four years – Congresses play a proper role in that or (almost equivalently) the need to organ- ise great and expensive international meetings every two years. The arguments (presented here in simplified form) were of Fourth Diderot Mathematical Forum two kinds: (1) real scientific gain is now obtained Mathematics and Music mostly by participating in specialised L. Mazliak (Paris) conferences on specific topics, rather than in ‘everything-touching’ big con- On 3-4 December 1999 the Fourth Diderot Mathematical Forum, on ferences; Mathematics and Music, will be held in Lisbon, Paris and Vienna. As for all (2) great meetings (congresses) are expen- events in this series, each city has selected a specific subtheme on which the sive, and the ratio of scientific gain to meeting will focus. These subthemes are: cost seems to be too small. Lisbon: A historical study of the connection between the two domains; Radical options presented during this dis- Paris: The problems around the formal systems for composition in the 20th cussion (and repeated a few times since century; then) described huge congresses as ‘relics Vienna: The mathematics of sound. of the past’. Such radical opinions were A round table between the three sites is scheduled for Friday afternoon on the (and are) not shared by many mathemati- theme: Is the link between mathematics and music a cultural or a natural one? Details cians, but they should be noted. of the programmes and local arrangements can be found on the web site: Presenting the opposite point of view, www.emis.de, with links to other sites. we notice that there is a growing trend The connections between mathematics and music are among the classical towards specialisation and a tendency to themes studied by philosophers. One of the main sources for this interest can organise specialised conferences and so- be found in the Pythagorean system that connects with elementary arithmetic called workshops, and it is thus reasonable the fundamental components of sounds (such as those emitted by a vibrating and necessary to bring together mathe- string) and all parts of the universe such as planets and stars. For centuries this maticians working in distinct areas of vision of the world, inherited from the ancient Greeks, was spread by the mathematics and give them opportunities Church-dominated Western scientific culture. It had the great advantage of to exchange ideas and enable interdiscipli- providing a unified system of the World, ruled by simple laws that could be nary discussions. It is also important to regarded as a proof of God’s rationality; all the teaching of medieval Christian build interpersonal relations, so that math- science could be summarised as ‘Reason is God’s part in Man’. With the help ematicians can meet each other as friends of reasoning, Man could tear the hidden order of Creation from the appearing and not only as scientific partners. chaos of the world. There are other important, and proba- On the beautiful allegorical rose of Laon cathedral in the north of France, bly deeper, reasons that make large and the artists of the Middle Ages have represented theology surrounded by its regular congresses very fruitful. Each such court: arithmetic, geometry and music are members of it. However, early in congress summarises (in some sense) what history (even in Greek times), serious fractures began to appear in a system that has been achieved in mathematics during was too perfect and too rigid: slowly, but inevitably, mathematics and music the last few years and identifies the most have followed separate ways. However, the weight of the cultural heritage was important results obtained since the pre- so great that its effects were still present many centuries after musicians had ceding congress. This is realised by the ceased to have a strong connection with pure science. It is impressive that great prestigious prizes awarded to the authors scientists, such as Kepler, Euler and Lagrange, have been interested in music at of the most spectacular achievements and a scientific level and have looked for formal systems. by the selection of the invited talks. Lists Attempts to find unified scientific explanations for music in the 19th centu- of invited lectures (both plenary and sec- ry quickly miscarried: some of them were clever, but most were quite wretched, tional) indicate the most important fields – as described in Fichet’s ‘Musical theories of music in the XIXth and XXth cen- not necessarily those with the most impor- turies’. In a sense, the twelve-tone system of Schönberg and the Vienna school tant and spectacular results! Thus, the may be seen as another attempt to obtain a unified system; when reading major international and prestigious con- Webern’s cycle of conferences in 1932 (Path to the new music), it is surprising gresses both summarise the past and stim- how much energy the author spends in ‘proving’ (through a careful choice of ulate the future, at least until the next con- examples!) that the twelve-tone system embodies all the previous music. gress. The attempt was doomed to failure. However, the twelve-tone system Accepting the above arguments, one marked a real turn in modern music: as an ‘official’ ending of tonality, it might say that the International opened the door to the idea that composers were free to choose their own com- Congresses of Mathematicians are suffi- positional systems. And here, mathematics returned to offer a platform on cient and that the European Congresses which artists may base their personal language. Also, new importance was given should be abolished. However, analysing to the mathematical study of the particular physical phenomenon we call ‘musi- the results and opinions from the first two cal sound’, in order to obtain a better understanding of it and to develop new European Congresses, we can see that they technology (such as new instruments) that can be used as a resource for com- were well placed between the ICMs, and posers. that their format ensures that the above At the turn of the 21st century, the Diderot Forum can be seen as an attempt aspirations are realised and help to moti- to take stock of the millenial history of the lively connections between the two vate the next congress. We also note that domains. the European Congresses do not duplicate the ICMs; in particular, the European 4 EMS September 1999 EMS NEWS Introducing the Editorial Team : part 1
Robin Wilson (Editor-in-Chief) is a Senior Lecturer in Mathematics at the Open University, UK, where he has been since 1972. He is also a Fellow of Keble College, Oxford University, and a frequent Visiting Professor at Colorado College, USA. He received his Ph.D. from the University of Pennsylvania in 1968 for a thesis in number the- ory. His mathematical interests lie mainly in graph theory and the history of mathematics. In the former he has been particularly involved with graph colourings and in the latter his concerns are mainly British mathematics, particularly of the late 19th and early 20th cen- turies, and the history of combinatorics. He has written and edited over twenty books in these areas, and won the Mathematical Association of America’s Lester Ford award for outstanding expository writing. He has served on the British Combinatorial Committee and the committee of the British Society for the History of Mathematics. He is actively involved with the popularisation of mathematics and enjoys lecturing to school and college students and to adults interested in mathematics.
Krzysztof Ciesielski (Associate editor and Societies editor) works in the Mathematics Institute of the Jagiellonian University, Kraków, from which he obtained his Ph.D. in 1986. His mathematical interests include dynamical systems, topology and analysis. He has been a Correspondent of The Mathematical Intelligencer since 1987. His wife Danuta is also a mathematician, who is interested in classical geometry and complex analytic geometry. He and his friend Zdzislaw Pogoda have co-authored about 100 articles presenting mathemat- ics to a general audience. They have been given several awards, including the Dickstein Prize (1995) which is given once every few years by the Polish Mathematical Society for an outstanding contribution to mathematical culture; they are the youngest ever recipients of this prize. They have written two popular books: Boundlessness of Mathematical Imagination (1995) and Mathematical Diamonds (1997). Both of these books were best-sellers in Poland.
Steen Markvorsen (Associate editor and Interviews editor) received his Ph.D. from the Technical University of Denmark in 1983. His main mathematical interest is in differ- ential (and distance-) geometry. In particular he is concerned with the geometric syn- thesis between curvature, form and function, including its applications to a spectrum of topics ranging from general relativity to biology. Together with a group of researchers he is currently exploring the role and potential of computer experimentation in mathe- matical research, in particular within the area of curvature geometry. He has been a member of the board of the Danish Mathematical Society and is actively promoting and disseminating mathematics inside and outside university circles.
Kathleen Quinn (Associate editor and Conferences editor) is a Lecturer in Mathematics at the Open University, UK. She obtained her Ph.D. in 1991 from London University. Her research interests lie mainly in design theory and the applications of combinatorics to cryptography. She was a lectur- er in London (at the Roehampton Institute) from 1991 to 1995, and then a research fellow at the Open University for four years.
June Barrow-Green (Anniversaries editor) is a Research Fellow in the History of Mathematics at the Open University, UK. She graduated from King’s College, London, and received her doctorate from the Open University. Her research interests include 19th- and 20th-century British and European mathematics (in particular, the work of Henri Poincaré). She is also working on the use of databases and the use of the World Wide Web as research tools in the history of mathematics. A former secretary of the British Society for the History of Mathematics (BSHM), she is currently a member of the BSHM Council. EMS September 1999 5 EMS NEWS tions. They are: G. Anichini, G. Bolondi, The Council may, at its meeting, add to B. Branner, J.-M. Deshouillers, K. the nominations received and set up a MeetingMeeting Habetha, M. Karoubi, T. Kuusalo, A. Nominations Committee, disjoint from the Lahtinen, L. Màrki, R. Piccinini, and D. Executive Committee, to consider all can- Puppe. Of the eleven, B. Branner, J.-M. didates. After hearing the report by the ofof thethe Deshouillers and M. Karoubi cannot be re- Chair of the Nominations Committee (if elected because they have served in this one has been set up), the Council will pro- EMSEMS CouncilCouncil capacity for eight years. ceed to the elections to the Executive Nomination papers for these elections Committee posts. 7-8 July 2000, Barcelona will appear in the December issue of the Delegates to the Council meeting, who First announcement Newsletter. Six delegates were elected for are planning to attend the European the term 1998-2001, so they will continue Congress of Mathematics, are advised that The EMS Council meets every second year. unless they inform the Secretariat to the their accommodation arrangements The next meeting will be held in Barcelona contrary by 31 December 1999. should be made through the ECM. For on 7-8 July 2000, before the 3rd European The Executive Committee is responsi- delegates to the Council who are not Congress of Mathematics. The exact loca- ble for preparing the matters to be dis- attending the ECM, an address for accom- tion will be announced later. cussed at Council meetings. Items for the modation arrangements will be provided Delegates to the Council will be elected agenda of this meeting of the Council later. by the following categories of members, as should be sent as soon as possible – and no per the Statutes. later than 10 March 2000 – to the EMS Secretariat: Ms. Tuulikki Mäkeläinen (a) Full Members: Full Members are nation- Secretariat in Helsinki. Department of Mathematics al mathematical societies, which elect 1, 2 The Council is responsible for electing P.O. Box 4, FIN-00014 or 3 delegates according to their size and the President, Vice-Presidents, Secretary, University of Helsinki, Finland resources. Each society is responsible for Treasurer and other members of the David Brannan the election of its delegates. Each society Executive Committee. The present mem- Secretary of the EMS should notify the Secretariat of the EMS in bership of the Executive Committee, Helsinki of the names and addresses of its together with their individual terms of Annex : Timetable for the delegate(s) no later than 10 March 2000. office, is as follows. As of 1 July 1999, there were 47 such soci- President Council Meeting eties – which could designate a maximum Professor Rolf Jeltsch (1999-2002) September 1999: Letters are sent to full, of 69 delegates. Vice-Presidents associate and institutional members as well (b) Associate Members: There are two associ- Professor Andrzej Pelczar (1997-2000) as delegates giving information on the ate members, namely the Gesellschaft für Professor Luc Lemaire (1999-2002) Council meeting. Specifically, points for Mathematische Forschung and the Secretary the agenda and suggestions for future European Mathematical Trust. Their cur- Professor David Brannan (1999-2002) members of the Executive Committee are rent common delegate is elected until Treasurer invited. (Delegates are kindly requested to 1999, so their delegate has to be elected in Professor Olli Martio (1999-2002) keep the Secretariat informed of their cor- 2000. According to the Statutes, ‘delegates Members rect and up-to-date addresses.) representing associate members shall be Professor Bodil Branner (1997-2000) 1 November 1999: Following the by-laws, elected by a ballot organised by the Professor Marta Sanz-Solé (1997-2000) the number of individual members is Executive Committee from a list of candi- Professor Anatoly Vershik (1997-2000) recorded to determine the number of their dates who have been nominated and sec- Professor Doina Cioranescu (1999-2002) delegates. onded, and have agreed to serve.’ Professor Renzo Piccinini (1999-2002) (c) Institutional Members: There are three December 1999 Newsletter: Information on the Council meeting is printed again. A institutional members, Institut Non- Under Article 7 of the Statutes, members nominating slip for the delegates of the Lineare de Nice, the Moldovian Academy of the Executive Committee shall be elect- individual members is given and sugges- of Sciences and the Mathematical Institute ed for a period of four years. Committee tions for Executive Committee members of the Serbian Academy of Sciences and members may be re-elected, provided that are invited. Arts. Their common delegate is elected till consecutive service shall not exceed eight 31 January 2000: Deadline for nomina- 1999, so their delegate has to be elected in years. Andrzej Pelczar has served on the tions for delegates of individual members. 2000. According to the Statutes, ‘delegates Executive Committee for eight years, so he February 2000: The ballots for delegates representing institutional members shall cannot be re-elected. of individual members are sent to individ- be elected by a ballot organised by the It would be convenient if potential nom- ual members. Executive Committee from a list of candi- inations for office in the Executive March 2000 Newsletter: Candidates for dates who have been nominated and sec- Committee, duly signed and seconded, delegates of individual members are onded, and have agreed to serve.’ could reach the Secretariat by 10 March announced. The venue and meeting times (d) Individual Members: A person becomes 2000. It is strongly recommended that a of the Council meeting are repeated. an individual member either through a statement of intention or policy is enclosed April 2000: A letter is sent to each dele- corporate member, by paying an extra fee, with each nomination. If the nomination gate, containing the agenda of the Council or by direct membership. On 30 June comes from the floor during the Council meeting. 1999, there were some 1900 individual meeting there must be a written declara- June 2000 Newsletter: The results of the members and, according to our statutes, tion of the willingness of the person to elections for delegates of individual mem- these members will be represented by 19- serve, or his/her oral statement must be bers are announced. The venue, the meet- 20 delegates. The final count of individual secured by the Chair of the Nominating ing times, and the agenda of the Council members for these elections will be made Committee (if there is such) or by the meeting are given. on 1 November 1999. President. It is recommended that a state- June 2000: Material for the Council meet- The mandates of 11 of the present 17 ment of policy of the candidates nominat- ing is sent to the delegates. delegates end on 31 December 1999, and ed from the floor should be available. so elections must be held for their posi- 6 EMS September 1999 NEWS OxforOxfordd doctoratedoctorate forfor AndrAndrewew WWilesiles In June Andrew Wiles received an honorary Paraphrase in English doctorate from his alma mater, Oxford I do not imagine that there is anybody in this University. For those readers who wish to try learned company who could not recite in uni- out their Latin, the citation by the Public son with me the formula: There is no whole- Orator Jasper Griffin was as follows; a trans- number solution to the equation xn + yn = zn, lation appears below. We thank Jasper Griffin where n is greater than 2. That is a very new solutions; especially as there are said to be for permission to reprint his citation. state of things. Until recently most of us only fifty people in the world who fully would have agreed with Cicero, who said that understand it, and I freely confess that I am Professor ANDREW WILES, FRS, Eugene mathematicians concern themselves with a not one of them. There was a time when peo- Higgins Professor of Mathematics, Princeton subject matter which is not only various and ple were inclined to criticise him for over- University. rarified but also obscure; but now discoveries confidence, in taking on such a problem sin- Neminem, credo, in hac hominum litteratissimo- in mathematics appeal to the ears of the gle-handed; there was a time when he came rum frequentia reppereris quin mecum hanc for- unlearned, and quite ordinary people feel an close to despair, as the theorem, which had mulam conceptis verbis concinere possit: Nullam in interest, even if not a well informed one, in its appeared to be defeated, suddenly put forth infinitum ultra quadratum potestatem in duas most abstruse areas. Anyone who is interest- new and unexpected difficulties. But all was eiusdem nominis fas esse dividere. Haud ita pri- ed in becoming famous should consider the well: his intellectual power prevailed, like an dem res aliter se habebat, nos plerique Ciceroni ipsi career of Professor Andrew Wiles and think Oedipus he solved the riddle of the Sphinx. adsensi eos dixissemus qui mathematici vocentur twice about passing up a mathematical His achievement has been greeted with uni- non solum recondita in arte et multiplici subtilique career. After spending many years at number versal applause by the mathematical commu- sed etiam in magna rerum obscuritate versari; sed theory, first here, then at Cambridge, and nity, although some of them view with regret hodie et Pythagorae arcana vulgi aures titillant, et most recently in Princeton, he has attained the disappearance of a venerable puzzle, and homines devia illa mathematicorum latibula visi- such celebrity that he has become recognis- the loss of the mingled pleasure and pain of tant ita indocti ut nihilominus curiositate ducan- able to laymen and to those with no profes- inconclusive mathematical endeavour. tur. itaque quisquis famam sibi adpetit huius quem sional interest in the subject. At a very early I present the Archimedes of our time, the produco vitam contemplatus ne eruditum geome- age he was attracted by algebraic number the- outstanding master of numbers, the incom- trarum pulverum aspernetur. hic enim cum diu in ory and decided that he would try to produce parable unriddler of the Last Theorem, intimis rei algebraicae medullis habitarit, tam pri- a proof of the last theorem of Pierre Fermat, Professor Andrew Wiles, FRS, Honorary mum nobiscum quam postea apud Cantabrigienses that classic problem which over the last three Fellow of Merton College, for admission to et postremo apud Princetonienses, tam insignem hundred years had been attempted without the honorary degree of Doctor of Science. denique consecutus est gloriam ut etiam ab insciis success by so many eminent mathematicians. neque isti studiorum generi adscriptis agnoscatur. In his maturity he crowned his many other Admission by the Chancellor qui usque a primis aetatis suae annis austera achievements by producing a definitive You are a prince among mathematicians: by numerorum scientia delectatus notissimum illud proof, by means of procedures of extraordi- your intellectual power you have solved a Petri Fermati theorema, a tot tamque nobilibus nary subtlety and intellectual range. It would most intractable problem where others have mathematicis CCC ferme annos frustra temptatum, take far too long if I were to try to explain his failed. Acting on my own authority and that sibi proposuit probandum; quod quidem firmata achievement in detail, with its perplexing dif- of the whole University, I admit you to the iam aetate post alia egregia facinora rationibus ficulties and Professor Wiles’s most ingenious honorary degree of Doctor of Science. exquisitissimis summoque acuminis firmamento usus tandem firmissime elaboravit. longum sit si spinosissimas istius incepti difficultates, ingeniosis- sima huius artificia, singillatim percensere coner, European Women in Mathematics praesertim cum L, tantum homines esse dicantur qui quantum hic perfecerit animo recte aestimare The video Women and Mathematics across Cultures, is available from the EWM office in possint, quorum in numero me non esse confiteor Helsinki. It was shown at the ICM98 in Berlin after the panel on women and math- atque concedo. erant homines quibus hic nimis ematics. audax videbatur, qui tantae claritudinis problema The video explores the impact of cultural differences of the female condition allow- solus aggrederetur; erat tempus quo ipse paene des- ing four women mathematicians who have studied and worked in Europe and North perarat, cum theorema illud, devictum iam, ut and South America to tell their stories. Following a five-minute introduction to EWM, videbatur, atque superatum, tamen tanquam including some surprising statistics about women mathematicians in Europe, the four Hydra illa Lernaea insperatas difficultates subito women recount their personal experiences. The length of the video is 25 minutes, protulit atque produxit. sed res bene vortit: vicit and was directed by Marjatta Näätänen in collaboration with Bodil Branner, Kari Hag tandem vivida vis animi, invenit Sphinx and Caroline Series in 1996. Oedipodem suum, cui ita plauserunt universi The cassettes are in VHS and the prices are as follows: mathematici ut dolerent quidam sibi ademptas esse PAL SECAM/NTSC haud ingratas frustra ratiocinandi molestias, solu- in Europe 200 FIM 250 FIM tum denique esse venerabile istud aenigma. outside Europe 220 FIM 270 FIM Preasento temporum nostrorum Archimeden, (6 FIM is approximately 1 euro) numerorum magistrum singularem, theorematis SECAM and NTSC cassettes are not equipped with English subtitles, but the video is ultimi enodatorem incomparabilem, Andream Wiles, Societatis Regiae Sodalem, Collegi in English and a written text of the interviews is provided. Mertonensis Socium honoris causa ad scitum, ut Please send your order via e-mail to [email protected] stating which system admittatur honoris causa ad gradum Doctoris in you want. After we receive your payment, the cassette will be mailed to you. Please, Scientia. pay to ‘Euroopan naismatemaatikot’, account number 800017-702454141 with Leonia Bank plc, Helsinki, Finland, with swift code PSPBFIHH via Euro Netting Admission by the Chancellor System or via Eurogiro. The telex of Leonia Bank is 121 698. Personal checks cause Mathematicorum princeps ingeniosissime, qui us a lot of work and expense, so they cannot be used; neither can we accept credit quaestionem perdifficilem deficientibus ceteris vi cards. cogitationis devicisti, ego auctoritate mea et totius Mailing address: Riitta Ulmanen, Department of Mathematics, P.O.Box 4, FIN-00014 Universitatis admitto te ad gradum Doctoris in University of Helsinki, Finland. Scientia honoris causa. EMS September 1999 7 INTERVIEW Interview with Tim Gowers (Cambridge) Fields Medallist 1998 interviewer: Tom Körner You come from a distinguished intellectual idea I had of what research in different areas family with Cambridge connections? would be like when I made such an important Well, I would not put it that strongly, but my choice. But I was lucky and found myself in great-great-grandfather was a famous neuro- an area that suited me very well, and with an physiologist in his day, and his name is still excellent supervisor. very much known to some of my colleagues at Just as I was finishing my Ph.D., Pete Trinity College in that area. My great-grand- Casazza visited Cambridge for a year. I had father was a civil servant who is best known a bit of a lean patch for eighteen months or for editing Fowler’s Modern English Usage, so around that time, but had a boost to my and for a book he wrote called Plain Words. morale in the middle of it when, encouraged This was originally commissioned as a guide by him, I solved a problem in the finite- to good writing style for civil servants, but it dimensional theory. He used to carry around turned out to have a much broader appeal a list of unsolved problems in Banach space and is still in print today. My father is a com- theory, mostly infinite-dimensional, working poser, which is a highly cerebral occupation up enthusiasm for them. At the time, the Despite appearances, I haven’t exactly even if it is not usually classified as an intel- received wisdom amongst Banach space the- changed subject, because my Banach spaces lectual one. He and I are often struck by sim- orists was that the infinite-dimensional theo- results and what I have done more recently ilarities between his working methods when ry had stagnated, with the interesting prob- both fall under a general heading of what writing a piece of music and mine when tack- lems being inaccessible. He very definitely might be called applied combinatorics. That ling a complicated mathematics problem. felt otherwise, and had a surprisingly pre- said, I did make a fairly conscious decision to My father, grandfather and great-grandfa- scient view of how things might work out. apply what combinatorial skill I had in areas ther all studied at Cambridge, and so did sev- The next year I made my main infinite- other than Banach space theory, for several eral relatives on my mother’s side. Although dimensional breakthrough, solving the so- reasons. First, given that I have only one life, my parents lived in London, I boarded at called unconditional basic sequence problem, I don’t want to spend twenty years of it in a King’s College School, Cambridge, where I which was also solved independently by single area of mathematics, and I can’t was a chorister. Bernard Maurey, and for some time my understand those who do. Secondly, I felt research programme was obvious – many under a certain amount of external pressure I believe that Andrew Wiles also went to the questions in the infinite-dimensional theory to show that I could do more than churn out same school? (including several on Casazza’s list) suddenly results on Banach spaces (not that I myself Yes, ten years before me so we had teachers looked possible with the new techniques and regarded the process as churning out). in common. Mary Briggs, the wife of the ideas that had been introduced, and over the Thirdly, and most importantly, I always had a headmaster at the time, had graduated in next couple of years or so, many of them fell, side interest in combinatorics (not surprising- mathematics from Girton College, where she to me and to others. Towards the end I ly, given my research supervisor) and wanted was taught by Mary Cartwright, and during obtained my dichotomy result. This needed to solve one or more of the beautiful prob- my last year or so she gave me some individ- different techniques, based on results in lems that I had known about for years. ual teaching. I was very lucky to be taught by Ramsey theory, though the problem itself was My general approach to research is to try her, and by my later teachers as well. closely related to my other work. I was par- problems with a reputation for being diffi- Thanks to Mrs Briggs I got a scholarship ticularly pleased because it was a positive cult, but, in order to avoid the danger of wast- to Eton and there I had another inspirational result, and I had felt that I was getting a rep- ing years getting nowhere, I like to have sev- teacher, Norman Routledge, who had been a utation as a ‘counter-examples mathemati- eral on the go at once, spending a month or fellow of King’s. He did not allow himself to cian’. Incidentally, many of my counter- two here, a month or two there, until one of be limited to the syllabus but ranged far more examples have resulted from trying to prove them reveals a soft underbelly. This proce- widely. In my last two years at Eton, the positive results – it’s just that there’s not dure increases the chances of happy acci- mathematics specialists were given a weekly much you can do about it when one of these dents. For example, I built up my under- sheet of challenging problems which were turns out to be false. standing of infinite-dimensional Banach only loosely based on the syllabus, if at all. Of spaces by trying and failing to solve the so- course, boys being boys, we tended to do I think Maurey is the only mathematician with called distortion problem, later solved by nothing for five days and then rush at them whom you have collaborated to the extent of Odell and Schlumprecht. Without that, I for two days, but even so it was a very valuable writing a joint paper? would not have seen how to do the other experience. Such a thing was rare then and I Yes he is, and even that collaboration arose problems. am afraid it is even rarer now in the days of out of the accident of our solving the same More recently, I decided to think about school league tables and the like. problem at the same time. Although it was the Kakeya problem, starting completely Then I got into Trinity and fell under the initially a shock (for both of us) to discover from scratch, because I had been told that it spell of Dr Bollobás – another believer in that we had to share the result, which was was very important. I hadn’t been going long hard questions off the syllabus. So through- much more important than anything I had when I realised (what I now know is an out my education I always had strong and previously done, I am now very glad to have absolutely standard observation) that it was good influences. collaborated with Maurey, and proud of our basically a combinatorial problem about two joint papers. The second of these grew arithmetic progressions, which caused me to You had an excellent but not an outstanding out of the first, and I would definitely not think, by no means for the first time, about undergraduate career? have been able to do it on my own. Szemerédi’s theorem. I then had an idea for That would be a fair summary. In general, although I have nothing how to tackle the special case of arithmetic against joint work and sometimes resolve to progressions of length 3, and found that I How did you choose your subject for research? do more of it, I think I work more naturally had reinvented Roth’s original proof for this In my third year I did pure mathematics and on my own. When I discuss mathematics with case. However, this spurred me on to think I gradually narrowed down my preferences to other people, I often find that either I don’t about progressions of length 4, and a month analysis rather than algebra or geometry. understand what they are saying, or they pro- or two later I came up with a complete proof The course in Part III (the Cambridge fourth voke some idea in me which I want to think for the general case. I wrote it out as a fin- year) which I enjoyed most was in the geom- about alone and free of distractions. This is ished paper, with all the details. I then dis- etry of Banach spaces, given by Dr Bollobás, probably just a sign of inexperience on my covered that a seemingly unimportant lemma and the thought of being supervised by him part. was false, and the hole, like a hole in a piece appealed to me very much anyway. Looking of knitting, expanded to the point where I back it is amusing to remember how little How did you decide to change subject? realised that the main lemma on which my 8 EMS September 1999 INTERVIEW proof was based was also false. This was a them. Bollobás brings up his students not to Ball, David Larman, Ambrose Rogers – with crushing blow, but by now I was hooked on use notes, which I think improves lectures interests similar to mine. I used to commute the problem. Bollobás was very encouraging immeasurably. As he puts it, how can we from Cambridge, and found the train a con- at this point and said there must surely be expect our audience to learn several courses genial place to work, making at least one gen- something in the ideas I had had. In the end for an examination if we, who are supposed uine breakthrough on it. I managed, after a big struggle, to get a new to be experts, cannot even learn a small frac- proof of the general result. I don’t think I tion of one? It is hard work, but there is When did you first think that you might win a would have had the stamina for it if it had not nothing quite like the feeling of having suc- Fields medal? been for the earlier disappointment. The cessfully given a complicated lecture from Initially, I assumed that it was impossible to whole thing occupied me for two years and memory. get one for work in Banach spaces, which at has led on to another clear research pro- Teaching is of course very important, least saved me the bother of thinking about gramme which should keep me going for since our future audience depends on it. It it. However, about two years before the last some time. Meanwhile, I still have no idea irritates me that many writers seem to treat International Congress I started getting mys- how to solve the Kakeya problem. books and papers more as an opportunity to terious e-mail messages asking for lists of display their own knowledge and achieve- publications, copies of papers, and so on. Does it bother you that the things you work on ment than as a genuine attempt to convey it The messages were usually labelled urgent, are unfashionable? to their readers – that is, to teach it. In par- but did not explain why they were urgent. Not from a personal point of view, because it ticular, it is very common for proofs to be Even then, it was a long time before I dared has not impeded my career, but I have been presented over-neatly, so that, although one to wonder whether I was being considered. lucky. There are others in my sort of area can see that the steps are correct, one has to who are bothered, and with some justifica- work much harder to understand how any- When did you realise that you wanted to be a tion. I do think it is good to have to work body could have thought of them. Just a few mathematician? hard to interest other mathematicians – the well-placed remarks can make an enormous It was a gradual process. It was the subject I situation becomes unhealthy when it is either difference (“Actually, this is a natural thing to enjoyed most at school, partly because I impossible to do so because one’s area is com- do because ...”) but they are surprisingly rare. found it easiest. The system laid out a series pletely out of favour, or too easy because it is Many papers, or chapters in books, could do of hurdles in one’s path and to some extent I the latest craze. with much longer introductions, setting the just found that I had jumped the hurdles The unfashionability of combinatorics is scene, explaining why the results are interest- rather than being forced to find another job. partly a result of the familiar contrast ing and explaining the difficulties to be over- I became certain that I would like to be a pro- between elementary methods and big come. But I am not the first to air these com- fessional mathematician some time when I machinery. Twentieth-century mathematics plaints. I firmly believe that one should aim was an undergraduate – though, even then, I has seen many triumphs of the latter, and this not just at a specialist audience, but also at had little idea of what this meant. has naturally influenced people’s opinions. I those who would prefer to skim a paper and think also that if you get used to big machin- get some idea of what is happening without And when did you feel that you had become a ery, then problems which can be stated to a worrying about the technical details. mathematician? schoolchild begin to seem babyish. The stan- Several times. At almost every hurdle, in fact. dard answer of a combinatorialist to such an We have tended to talk about Cambridge. When I became a research student. When I attitude would be “Why not have a go at solv- What about your time elsewhere? got a research fellowship. But when I solved ing one?”, but this does not convince every- Although my heart has always been in the unconditional basic sequence problem, body. Cambridge, I had a very happy and produc- which was a problem with a name, which oth- Combinatorics has a reputation for being tive four years at University College London, ers had tried, then I felt for the first time that rather isolated, with few applications to the and am glad to have experienced life else- I had truly fulfilled a boyhood ambition. rest of mathematics, or at least important where (though never a life without cloisters). mathematics. I think this view rests on a mis- The department suited me well, as there were We thank the London Mathematical Society for permission to apprehension. Combinatorics does not con- several people there – David Preiss, Keith reproduce these photographs tain many powerful and difficult general results like the Atiyah-Singer index theorem, which can be directly applied over and over again. However, over the years, combinatori- alists have built up a considerable expertise in solving certain types of problems which would have been hopeless a generation ago and which can be used in many external fields. Although progress in mathematics can be a result of truly understanding difficult concepts until the proofs write themselves, I believe that it is not sufficiently recognised that many problems are at heart combinator- ial, and in the end if you want to solve them you simply have to get your hands dirty.
What is your attitude to teaching? I enjoy teaching, not all the time of course, but I find it sufficiently rewarding to wish to go on doing it. I enjoy the process of work- ing out why something that I understand causes difficulty to a student. (Unfortunately, the answer is often boring – they just haven’t learnt the relevant definitions or something like that.) I am fortunate enough to supervise a number of the very best undergraduates in the country, who are sometimes better than I am at what I am supposed to be teaching them. I find that lecturing can be directly benefi- cial to my research, because I have to under- stand even quite elementary material much Seven Fields Medallists at the opening of De Morgan House, the new headquarters of the London better if I am going to stand in front of two Mathematical Society (see EMS Newsletter 31). From left to right: Tim Gowers, Klaus Roth, Simon hundred intelligent people and explain it to Donaldson, Sir Michael Atiyah, Alan Baker, Richard Borcherds and Daniel Quillen. EMS September 1999 9 ICIAM99 (SMAI), the Sociedad Espanola de important contributions to the mathemati- Mathematica Aplicada (SEMA) and the cal theory of fluid motion, solid structure, ICIAM 99 Societa Italiana di Mathematica Applicata non-linear waves, scaling and asymptotics. e Industriale (SIAMAI). He constructed a deep connection between The Collatz prize 1999, awarded to a non-linear waves and general scaling argu- scientist under 42 years of age, went to ments. Among the many applications of Stefan Mueller for his highly original and this highly original and amazing theory profound contributions to applied mathe- are the scaling of turbulence, the analysis matics, the calculus of variations and non- of failure in solids, the dynamics of reser- linear partial differential equations, the voirs and the analysis of stratification in mechanics of continua, and mathematical geophysical fluid mechanics. The Maxwell material sciences. Mueller, born in 1962, prize is sponsored by the Institute of studied mathematics and physics in Bonn, Mathematics and its Applications (IMA) Edinburgh and Paris. He became full pro- and the James Clerk Maxwell Foundation, fessor at the University of Freiburg at the to provide international recognition to a age of 32, and shortly after, Vice-Director mathematician who has demonstrated of the famous mathematical Research originality in applied mathematics. in Edinburgh Centre at Oberwolfach. After a brief These prizes will be presented every appointment at ETH Zurich, he became four years at the ICIAM. Clearly, more 5 – 9 July 1999 one of the three directors of the Max than through their official descriptions, Planck Institute for Mathematics in the the world standing of their first winners Rolf Jeltsch Sciences in Leipzig in 1996. Stefan will set the level of achievement for future Mueller is one of the very few young math- winners. The presentation of these prizes (President of EMS) ematicians in the world who combine high- concluded the opening ceremony. quality mathematical skills with a feeling The conference was one of superlatives More than 1500 applied mathematicians for real world problems. The Collatz prize – and not only with respect to these prizes gathered from all over the world for a week was sponsored by the Gesellschaft für and their winners. In one week there were in Edinburgh. They came to attend the 4th Angewandte Mathematik und Mechanik 33 plenary speakers and about 210 mini- International Congress on Industrial and (GAMM). symposia (usually consisting of 3-4 lec- Applied Mathematics. ICIAM99 followed the The CICIAM Pioneer prize was awarded tures, but some running up to 26 lectures), traditions of its successful predecessors in to Ronald R. Coifman of Yale University and and about 1100 contributed papers were Paris (1987), Washington (1995) and Helmut Neunzert of the University of presented. Everything ran in parallel, Hamburg (1995). Kaiserslautern for very different contribu- even the plenary sessions, and sometimes The opening ceremony was held in the tions to applied mathematics. Coifman 28 mini-symposia were running simultane- beautiful McEwan Hall, a semicircular was honoured for his pioneering work in ously, so you cannot expect your corre- arena with fantastic paintings on the ceil- exploiting harmonics, and especially spondent to tell you all the highlights; it ing of the half dome. Unfortunately, wavelet analysis, to provide computational was simply impossible to attend and absorb H.R.H. Prince Philip, Duke of Edinburgh, methods and algorithms in a wide variety all that was presented. The few lectures I a Joint Patron of the Congress, couldn’t of important contexts involving signal and did attend were selected by personal inter- attend the opening as he was attending a image processing. Applications have est, curiosity, or just by chance. But the ceremony to honour the late King Hussein included FBI data files for fingerprints and proportion of excellent lectures was of Jordan. Instead, a message from him many other problems involving compres- extremely high. The two Schlumberger was read and he was able to join the con- sion and/or restoration of images and lectures I attended by F. P. Kelly on gress dinner in the Playfair Library at the sound. Who has not seen, when clicking Mathematical modelling of the internet and S. Old College of the University of on a webpage, how the images on the page Popescu on What is quantum computation? Edinburgh. Sir Michael Atiyah, Chair of are built up from large scale wavelets to were real eye-openers for me. Margaret ICIAM99, responded to the opening fine scales in seconds? Neunzert was hon- Wright’s lecture on What, if anything, is new speech and Lord Sainsbury, UK Minister oured for his work over the last twenty in optimization? and James A. Sethians’ pre- of State responsible for Science, welcomed years in developing ‘technomathematics’, sentation on Fast marching methods and level the delegates. The Celtic Brass opened the both as a scientific discipline and as a cur- set methods: evolving interfaces in fluid ceremony which announced the four new riculum now offered at more than twenty- mechanics, computational geometry and materi- CICIAM prizes, created since ICIAM95 in five universities, and in developing the al sciences really surpassed their usual Hamburg. The president of CICIAM, specialisation of industrial mathematics (already fascinating) performances. Reinhard Mennicken, presented the prizes through active consulting and modelling, Among the mini-symposia, I would like to to the winners. playing a leading role in the European single out the one on Computational science Jacques-Louis Lions was awarded the Consortium for Mathematics in Industry, and engineering: How to organize? How to Lagrange prize in recognition of his and for founding and directing the teach?, because this new curriculum will exceptional contributions to applied and Frauenhofer Institute for Techno- and have to be discussed in the near future by industrial mathematics throughout his Econo-mathematics at Kaiserslautern. many mathematics departments. career. He was cited as one of the most [You can read more about Helmut For those with insatiable appetites who distinguished and influential scientists of Neunzert in an interview he gave to Heinz found the presented material insufficient, this century in the domain of applied and Engl in the June 1999 issue of the EMS there were additional scientific events like industrial mathematics. A few of his out- Newsletter.] The Pioneer prize is funded by the Symposium on mathematics and the law standing contributions to our science and the Society for Industrial and Applied and a meeting on Maxwellian themes with his famous books were mentioned. Lions Mathematics (SIAM), and is given for pio- topics like ‘Testing Einstein in space: a founded and developed an important neering work introducing applied mathe- marriage of physics and technology’ or ‘A school of applied mathematics in France matical methods and scientific computing Maxwellian approach to modern cosmolo- which has had a strong influence in many techniques to an industrial problem area gy’. ICIAM99 really covered all the fields other countries. He has participated in or a new scientific field of applications. one could imagine – and even ones one many industrial programmes. With the Grigory Issakovic Barenblatt was awarded couldn’t. choice of Jacques-Louis Lions the prize the CICIAM Maxwell prize in recognition Three additional prizes were awarded committee has set an extremely high stan- of outstanding originality in his work in in the McEwan Hall, between two morning dard for future winners. The Lagrange applied mathematics. He is one of the plenary lectures. prize is a gift of the Société des most distinguished Russian applied math- The Dahlquist prize, established by Mathématiques Appliquées et Industrielles ematicians, and is well known for his SIAM in 1995, was awarded to Linda 10 EMS September 1999 ICIAM99 mittees and the whole scientific pro- gramme of ICIAM99 on the webpage: http://www.ma.hw.ac.uk/iciam99/. Finally, I should add that the EMS was pre- sent at ICIAM99 with a booth, jointly with Zentralblatt MATH. It gave the Chair of the Applications Committee (Heinz Engl) and myself a good opportunity to meet mem- ber societies and their representatives as well as other societies; you will be reading about some of the results from these talks in future Newsletters. One effect can be noted in this issue: SIAM and EMS are planning a cooperation agreement and you will find an article by the SIAM presi- dent, Gil Strang below. A corresponding article by myself will appear in the September issue of SIAM News. On Saturday the Committee for International Conferences on Industrial and Applied Mathematics (CICAM) had a business meeting. It prepared the election of the new president, to be done by mail Jacques-Louis Lions, winner of the Lagrange prize, at the 1994 ICM. this autumn. It also established new statutes. We shall report on these new Petzold for her important contribution to National Physical Laboratory, and the developments in the next Newsletter. effective numerical methodology for dif- Numerical Algorithms Group (NAG). We look forward to the next ICIAM, to ferential equations, especially the analysis You can find more information on all be held from 7-11 July 2003, in Sydney, of methods for differential algebraic equa- prizes, the exact citations, the prize com- Australia. tions, the construction of effective tech- niques for their solution, and the integra- tion of these and other techniques into robust software, thus making possible the reliable solution of large classes of ordi- SIAM and EMS joint conference on nary and partial differential equations aris- ing from engineering and science applica- Computational Science tions. This prize is awarded on a biannual basis. Gil Strang, SIAM President Germund Dahlquist received the newly established Henrici prize. He was cited for I am writing to introduce SIAM, the Society for Industrial and Applied Mathematics. his outstanding research and leadership in This is an especially good time to add new connections between SIAM and the EMS, numerical analysis. Dahlquist created the because our two societies are planning a joint conference. It will be the first major fundamental concepts of stability, A-stabil- conference for SIAM in Europe and the first big conference on Computational sci- ity and the non-linear G-stability for the ence for the EMS. Rolf Jeltsch, EMS President, is preparing a similar column for numerical solution of ordinary differential SIAM News, to introduce the EMS to members of SIAM. equations. He succeeded, in an extraordi- I hope you have seen SIAM News, which goes to all members and is very widely nary way, in relating stability concepts to read – and above all the SIAM journals which are at the centre of our work. There accuracy and proved the deep results that are now eleven journals covering a wide range, from classical problems in analysis are nowadays called the first and second to discrete mathematics and optimisation. Maybe this is an important point about Dahlquist barriers. His interests, like SIAM – it has a very broad view of what applied mathematics really is. Henrici’s, are broad, and he has con- Another important point about SIAM is that it is truly international, as mathe- tributed significantly to many parts of matics is. More than a third of our members live and work outside North America. numerical analysis. As a human being and I am personally very happy about that, and am working in my two years as president scientist, he gives freely of his talent and to do everything possible for applied mathematics worldwide. So much of the best knowledge to others and will remain a research is joint work – we are already working together! I have had total encour- model for many generations to come. This agement in this cooperation from inside and outside SIAM. was the official citation; I wish to add that I Let me mention some new directions for the society? One is to establish an activ- myself benefited a lot from his generosity. ity group in the life sciences. Biology, medicine and pharmacology are areas of The Henrici prize is jointly sponsored by tremendous scientific growth, and the mathematical and computational parts are SIAM and The Swiss Federal Institute of increasingly important. The activity groups in SIAM are responsible for focussed Technology, ETH Zurich. conferences every two years and I believe that this new group will grow quickly. The Matteo Frigo and Steven Johnson of the environment is also a part of their interest. The linear algebra group has been par- Massachusetts Institute of Technology ticularly active in Europe, and I anticipate a new group in 2000 on imaging science. received the Wilkinson Prize for FFTW, May I call your attention to our web page: www.siam.org. It has much more infor- ‘the fastest Fourier transform in the West’, mation about the society than I can give here. You can see the calendar of confer- a C library for the computation of the dis- ences, and how to join SIAM, and the editorial boards of all the journals (which now crete Fourier transform that automatically come electronically before the printed editions). I am happy to give my own email tunes the computation for any particular ([email protected]) for anything I can do. hardware in order to produce efficient One more word about our joint conference. The first discussion was with Rolf code. The Wilkinson prize for numerical Jeltsch and Heinz Engl. The whole topic of computational science and engineering software was established in honour of the is coming into prominence for students too. The key is to know about the science outstanding contributions of James Hardy as well as the mathematics (and the computation). I am convinced that the confer- Wilkinson to the field of numerical soft- ence will be a success. ware. This prize is jointly sponsored by Argonne National Laboratory, the EMS September 1999 11 RESOURCES LOCATOR such a large database can have. But how is the ACM/UMRL database maintained? A universal ACM/UMRL for the organiser of a confer- A universal ence, colloquium or seminar series Here is the key concept: the ACM/UMRL database gets created and maintained automatically, using a web-engine that queries the web pages of all seminar series, mathematicalmathematical conferences or colloquia registered with ACM/UMRL. This requires the active cooperation of the organisers of these events, but that is straightforward: the URL’s of the announcement web pages rresouresourcesces have to be indicated to the ACM/UMRL- master and these pages must include some very simple tags that the ACM/UMRL engine can recognise and retrieve. Another advantage of this procedure is locator?locator? that the database always remains up-to- date. Detailed explanations are available Alain Damlamian on the ACM/UMRL server
12 EMS September 1999 ANNIVERSARIES 1999 Anniversaries
less well-known, part deals with directions defence were professors at Copenhagen Caspar Wessel in space. The treatise was presented at a University, two official (H.-G. Zeuthen and meeting on 10 March 1797 to the Academy Julius Petersen) and one from the audi- on representing complex and accepted for publication. From the ence (T.-N. Thiele). Immediately after, numbers (1799) start (1742) the Academy had decided to Christian Juel, docent at the Technical Bodil Branner publish articles written by members in University, called attention to Wessel’s Danish, not Latin. The statute of the achievement by publishing a short paper The Norwegian surveyor Caspar Wessel is Academy was relaxed in 1796 to allow non- (in Danish) about it and Sophus Lie now recognised as the first to have given a members to submit articles and Wessel’s reprinted Wessel’s paper in the Archiv for geometrical interpretation of complex treatise became the first of this kind to be Mathematik og Naturvidenskab in 1895. The numbers and their rules of composition. published. Academy honoured Wessel by publishing a This year we celebrate the bicentenary of At the end of the 18th century there was French translation of his treatise by H.-G. his paper in Det Kongelige Danske nobody around who could understand or Zeuthen in 1897. Last year the Academy Videnskabernes Selskabs Skrifter, Nye Samling, appreciate the scope of Wessel’s work. If held a Wessel Symposium organised by V, Kiøbenhavn, 1799, pp. 469-518 [the col- there had been, the paper would probably Jesper Lützen, and this year a complete lected papers of The Royal Danish have been translated into Latin or German English translation of Wessel’s treatise will Academy of Sciences and Letters]. as other important publications were. appear (for the first time) in the proceed- Although his work was ignored and had no There were many people of German origin ings [1], together with a Wessel biography influence on later mathematical develop- in Copenhagen at the time, such as J. N. and a paper on the history of complex ments, his story is worth telling. Tetens, who had come to Copenhagen numbers. In 1796 Wessel completed the triangu- from a professorship at Kiel University in Caspar Wessel was born in 1745 in lation of Denmark and Schleswig that, philosophy and mathematics. He encour- Vestby in Norway, south-east of Christiania together with astronomical observations, aged Wessel to write his results down and, (now Oslo). He was the sixth child (of 14) formed the basis of the first real cartogra- as the leader of the mathematical class of of the curate Jonas Wessel and his wife phy of Denmark; this work was done under the Academy, he presented Wessel’s work Helene Marie Schumacher. When he was the auspices of the Academy. In the same at the meeting. Wessel himself was not pre- 12 years old he was sent to the Cathedral year Wessel wrote a mathematical treatise sent. School in Christiania with two of his elder entitled Om Directionens analytiske Soon after, Wessel’s treatise seems to brothers. There was no university in Betegning, et Forsøg anvendt fornemmelig til have been completely forgotten. It was Norway, so the three brothers moved to plane og sphæriske Polygoners Opløsning (On rediscovered about a century later when Copenhagen, the capital of the dual the analytical representation of directions; Sophus A. Christensen mentioned it in his monarchy Denmark-Norway, to study at an attempt applied chiefly to solving plane doctoral thesis (On the development of mathe- the university there, the elder brothers and spherical polygons). Its first part deals matics in Denmark and Norway in the 18th Johan Herman and Ole Christopher in with directions in a plane, and the second, century). The three opponents at his 1761 and Caspar in 1763. They all had some financial support from home to start out with, but it soon became necessary to find extra income. Both Ole Christopher and Caspar became involved with survey- ing for the Academy. Ole Christopher earned his living this way while he was a student; after taking a degree in law in 1770 he embarked on a successful legal career, attaining one of the highest posi- tions in Norway. For Caspar surveying became an engagement for life although, like his brother, he completed a degree in law. Their brother Johan Herman Wessel became a poet. Although he was not very productive and died rather young, many of his poems are still remembered in Norway and Denmark; they have a special humour and are hard to translate. The first plan for a modern topograph- ical measurement of Denmark was pro- posed to the King in 1757 by a student, Peder de Koefoed. The plan was approved and Koefoed started to work, but unfortu- nately he died three years later having accomplished little. But the time was ripe and the way prepared for a more ambitious plan put to the Academy by Christen Hee, professor in mathematics, and Thomas Bugge (1740-1815), Koefoed’s assistant. The plan involved both topographical sur- veying and the method of triangulation to determine geographical coordinates. This long-term project of great national (and international) interest became essential for the development and strengthening of the Title page of Wessel’s 1799 treatise. young Academy. EMS September 1999 13 ANNIVERSARIES
Page 50 of Wessel’s 1787 surveying report, Trigonometriske Beregninger; in Section 5 he expresses the direction of the nth tangent vector in Fig. 3 by Tn.(cos wn + √-1. sin wn). 14 EMS September 1999 ANNIVERSARIES calculated. Angles were measured by the so-called geographical instrument or Ekström’s circle. For trigonometrical surveying the instrument was used with the circle in hor- izontal position and placed above one trigonometrical point of a triangle, with one telescope pointing towards one of the other points in the chosen triangle and the other telescope in the direction of the third point in the triangle. The instrument was also used for astronomical measurements with the circle in vertical position, measur- ing the height of the sun or certain stars. The origin of the triangular net had been chosen as the Royal Observatory in Copenhagen placed in the Round Tower. The building of the Round Tower was started in 1637 and the observatory was established by Longomontanus, a pupil and former assistant of Tycho Brahe (1546-1601). Brahe had been the first to use a triangular net in surveying, to deter- mine the position of his observatory Uraniborg on the island Ven relative to the coastlines of Sweden and Denmark. The method was later pioneered by the French Abbé Jean Picard who visited Denmark in 1671-72 in order to compare the longi- The geographical instrument shown in vertical position, from Thomas Bugge’s Beskrivelse over tudes of the new Paris Observatory and den Opmaalings Maade som er brugt ved de Danske geografiske Karter, Kiøbenhavn, 1779. Uraniborg and the Round Tower. When Hee and Bugge educated the two first trigonometrical surveying, to succeed the the trigonometrical surveying of Denmark surveyors, one of whom was Ole successor of Ole Christopher. Neverthe- started in 1762, that of France had already Christopher Wessel, and the work started less, Hee wrote a supporting letter: ‘None of been completed, forming the basis of the in 1762. Two years later Ole Christopher the surveyors has been more useful to us than he famous Cassini-maps. needed an assistant and chose Caspar. At has, during the summers he has been surveying From May 1782 to the summer of 1785 first Bugge was responsible for everything and in the winter time he has been working as a Wessel was again on leave from the including the trigonometrical measure- designator, which in the fourteen years he has Academy, but this time with its recommen- ments (the triangulation), the most theo- stayed with the surveying has ruined his health dation. He was made responsible for the retical part of the surveying, but Ole and been an obstacle to his studies in such a complete trigonometrical surveying of the Christopher Wessel soon took over this way, that if he once again has to interrupt his duchy Oldenburg West of Bremen. In a let- responsibility as well. However, his salary studies he is perdu and will never pick them up ter of recommendation, Bugge wrote in as an assistant was so low that Caspar could again. Last winter when he nolens volens had to 1781, ‘He possess a lot of theoretical knowledge not survive on it alone, and he asked the draw the general map of Zealand he was once of algebra, trigonometry and mathematical Academy for a rise if he took on the draw- more distracted in his studies, and then I geometry, and as far as the last point is con- ing of maps in addition to his surveying. promised him never again to disturb his circles.’ cerned, he has come up with some new and He was granted increased pay and made The sabbatical was granted and Wessel fin- beautiful solutions to the most difficult problems responsible for the construction, reduction ished his degree in law. No trigonometri- in geographical surveying’. The Oldenburg and drawing of maps based on the geo- cal surveying took place that year. area was later (around 1824) re-triangulat- graphical and trigonometrical surveying, From 1779 to 1796 he worked as a ed by Gauss; his instrument and part of his then presenting the model from which the trigonometrical surveyor, still spending his triangular net can be seen on German 10 coppersmith would make plates and pro- summers in the countryside measuring. DM notes. duce the final prints. A test map had During the winters he was occupied with In the 1779 surveying report Wessel already appeared, representing the trigonometrical calculations based on the explained how a map of an ellipsoid can be County of Copenhagen; officially it had collected data, judging the validity of the obtained by projecting points of the ellip- been drawn by Ole Christopher Wessel, measurements, and constructing and soid onto a cone and unfolding. The cone but from the surveying diary we know that drawing the resulting triangular maps. His he used for Denmark was formed by the Caspar had drawn part of it. reports also contained shorter articles tangents to the meridians through points From May to September or October, the describing the methods behind the mea- with the same latitude as the Observatory. surveying took place in the countryside surements and cartography. His work When he returned from Oldenburg he from early morning to late evening (weath- required both practical and theoretical elaborated on this method and in his 1787 er permitting) every day except Sundays. A skills, as well as accuracy, patience and a report he described how he came closer to fair copy of the surveying diary had to be breadth of outlook. the actual measurements by using several sent to the Academy, together with the The purpose of the trigonometrical sur- cones, each one over points of a fixed lati- topographical maps measured that sum- veying was to establish a network of tude; in the unfolding he pieced the dif- mer, marking the locations of towns, trigonometrical points, to determine their ferent cones together along tangents to a churches, castles, mills and woods, the triangular net, and to supplement this with common meridian. In his description of courses of roads and streams, and the posi- an astronomical determination of the lati- how to calculate the coordinates of a point tions of coastlines and islands. During the tude and longitude of some of the points in the unfolding, he expressed a direction winter there should then be time for study, and of the direction of the meridian in the plane by T(cos w + √-1.sin w). So at but for Caspar Wessel this proved difficult through such points. One distance, a base least as early as 1787 he had the geometric since he was then working on the reduction line, between two neighbouring points in interpretation of complex numbers as of maps. By 1778 he had became so des- the triangular net had to be measured with directions in a plane. There is no trace of perate that he asked for a sabbatical year great care; all angles of all triangles in the the product rule in this report. But know- with full salary. This was not popular since net were measured several times and from ing that he was a master in handling he had just accepted responsibility for the these data the rest of the distances were trigonometrical formulas and noticing how EMS September 1999 15 ANNIVERSARIES he wished to change from a direction given Brown) and eigenfunctions (Norrie by (cos wn + √-1.sin wn) to one given by Everitt). (cos wn+1 + √-1.sin wn+1) through a turn of the angle of (wn+1 - wn), it is possible that Early Years he obtained the geometrical interpretation The Titchmarsh family can be traced of the product rule by pursuing this fur- back many centuries, in the area to the ther. south-west of Cambridge. There is a village As already explained, he wrote his one called Titchmarsh in Northamptonshire – and only mathematical treatise in 1796. It but it was around Royston that E. C. was clearly inspired by his work as survey- Titchmarsh’s forebears lived as local or, but there is no reference to this, and traders. Titchmarsh’s grandfather ran a the results he obtained did not simplify the grocery shop, but Titchmarsh’s father trigonometrical calculations in the survey- entered the church and became a ing. In part two of his treatise he expressed Congregational Minister – first in directions in space by refering to two Newbury, Berkshire, and later in Sheffield orthogonal complex planes with the real (Yorkshire). axis in common. Although his notation Ted Titchmarsh was born in Newbury, may look rather clumsy and the results the second of four children. In Sheffield he seem to drown in pages of symbols, there is attended King Edward VII School, where an underlying simple and elegant geomet- he performed well in most subjects. He rical idea which enabled him to obtain a later wrote: ‘The first occasion on which I ‘universal formula’ from which he derived distinguished myself was when I was in one all the spherical trigonometrical formulas. of the fourth forms. The headmaster for some unknown reason made the whole One of Titchmarsh’s favourite occupations In 1805 Wessel resigned. But he still was playing criket worked for the Academy when they needed upper school do an arithmetic paper, the Ted Titchmarsh was to write: ‘I was howev- his help; for instance, in 1808 he drew a same for all forms. The mathematical spe- er principally influenced by G. H. Hardy. triangular map of the duchies Schleswig- cialists in the sixth form came out top, and From him I learned what mathematical Holstein and added some explanations in I came next . . . At this point one had to analysis is, and at his suggestion I devoted French, since the maps were requested by choose either classical or modern subjects: myself to research in pure mathematics.’ the French Emperor Napoleon and sent to I was put on the classical side. I learnt enough Latin to pass and enough Greek to Dépôt général de la Guerre in Paris. In Oxford in the 1920’s fail. It had become clear that mathematics 1815 he was made a knight of the His most important contemporary in was my real subject, and I began to spe- Dannebrog in recognition of his excep- the 1920s was indeed Hardy, who had left cialise in it.’ tional contribution to surveying. He died Cambridge in 1920 to succeed William In December 1916 Titchmarsh won the in 1818. Esson (former deputy to J. J. Sylvester) as Open Mathematical Scholarship to Balliol Acknowledgement: This article is based on Savilian Professor of Geometry in Oxford. College, Oxford, going up in October joint work with Nils Voje Johansen; our Hardy was the founder of the Oxford 1917 for a term. However, he then went Wessel biography in [1] contains a list of research school in analysis, a tradition that away for almost two years on war service, as the material used. Titchmarsh followed ten years later when Second Lieutenant in the Royal Engineers he succeeded Hardy as Savilian professor. (Signals), going to France from August Bibliography Other Oxford contemporaries included 1918. He became a dispatch rider, on 1. Caspar Wessel, On the analytical represen- Edwin Elliott, the first holder of the horse and then motorcycle. He returned to tation of direction. An attempt applied chiefly to Waynflete Chair in pure mathematics, who Oxford in October 1919, being tutored by solving plane and spherical polygons (translat- wrote important books on invariant theory J. W. Russell. Mary Cartwright has written: ed by Flemming Damhus). With introductory but had ‘no sympathy with foreign modern ‘At Russell’s first lecture the room was chapters by Bodil Branner, Nils Voje Johansen methods’; W. L. Ferrar later described packed to the doors, and Russell said: “Ah, and Kirsti Andersen (edited by Bodil Elliott: ‘a man who has written books which there’s my clever pupil Mr Titchmarsh – Branner and Jesper Lützen, C.A. Reitzel, have put the works of his rivals on the he knows it all, he can go away.” Russell 1999, ISBN 87-7304-298-6. bookshelves is the worst lecturer who ever dictated his lectures word for word and 2. Olaf Pedersen, Lovers of Learning. A picked up chalk’, Elliott retired in 1921, examples were handed out – and then, if History of the Royal Danish Academy of being replaced by Alfred Lee Dixon who necessary, solution to examples. Some of Sciences and Letters 1742-1992. made few mathematical innovations but Titchmarsh’s solutions replaced the official Munksgaard, Copenhagen, 1992, pp. 89- knew some ‘very pretty things in 19th-cen- ones.’ Certainly his student career was very 104 & 119-132. tury geometry’; Dixon, a keen sportsman, successful, and he gained a First Class was so strong that he could break a walnut Honours Degree and won the Junior and Bodil Branner teaches in the Department of in the crook of his arm. On the applied Senior mathematical scholarships. Later, Mathematics, Technical University of Denmark. side, the Sedleian Chair of Natural Philosophy was held for over forty years by Augustus Love, who worked in continuum mechanics and electrodynamics and wrote E.C. Titchmarsh (b. 1899) a classic book on elasticity. In addition, the Robin Wilson new Rouse Ball Chair of Mathematics was founded in 1928 and the first holder was E. On 1 June, around 100 people came to the A. Milne. Mathematical Institute in Oxford to com- But in pure mathematics, Hardy was the memorate the 100th anniversary of the main influence. He invited many foreign birth of Edward (‘Ted’) Charles mathematicians to Oxford, such as the Titchmarsh (1899-1963). This gathering, number-theorist Edmund Landau, the organised by David Edwards, included Russian emigré Abram Besicovitch, and many of his former colleagues and four Georg Pólya, the first Rockefeller Fellow. generations of his family, including two of Hardy’s weekly evening mathematical his three daughters. Talks were given on gatherings were of great interest to his life (Robin Wilson) and his work in the Titchmarsh, Mary Cartwright and others, areas of Fourier analysis (David Edwards), as were the meetings of the Oxford the Riemann zeta-function (Roger Heath- E.C. Titchmarsh as a young lecturer Mathematical and Physical Society, found- 16 EMS September 1999 ANNIVERSARIES associated with second-order differential equations. His method of writing them was interesting; after researching on a topic for a couple of years he would ‘sign off’ with a book representing the synthesis of his own discoveries. He also wrote a popular Mathematics for the General Reader. His achievements became widely recog- nised. He was President of the London Mathematical Society from 1945 to 1947 and received its highest honours – the De Morgan Medal and the Berwick Prize. Sheffield University awarded him an hon- orary doctorate in 1953 and he was a ple- nary speaker at the Amsterdam International Congress the next year. The Royal Society awarded him their Sylvester Hardy leads a criket team of Oxford mathematicians during a British Association meeting in 1926; Medal in 1955. Titchmarsh is fourth from the right. Hardy dubbed this photograph “Mathematicians v The Rest of the World”. On a personal basis, he was a man of few words, but his silences were ‘benevolent ed by Sylvester in 1888 and celebrating its he was elected a Fellow of the Royal and never oppressive’. While in Oxford he 200th meeting in 1925. Hardy’s influence Society, became President of the Liverpool became Curator of the Mathematical also extended beyond mathematics; his Mathematical Society, and wrote his first Institute, issuing keys to new graduate stu- regular cricket matches frequently book, a Cambridge tract on the Riemann dents. Sir Michael Atiyah recalls that on involved Titchmarsh, who had a passion zeta-function. arrival in Oxford as a graduate student ‘I for the game – indeed, his uncle was a pro- In 1931 Hardy returned to Cambridge was duly ushered into his big room, where fessional cricketer. to take up the Sadleirian Chair vacated on he was sitting at his desk. I sat down and he The 1920s were important years for the resignation of E. W. Hobson. By handed over the key, and I then expected Titchmarsh. In 1923 he was appointed to a chance, Titchmarsh was visiting Oxford to a speech of welcome or some words of Senior Lectureship at University College, examine a doctorate and bumped into advice, but we just sat in silence. After five London, giving undergraduate lectures, Ferrar who asked him whether he’d minutes I left.’ supervising postgraduates, and starting to applied for Hardy’s vacant Oxford Chair. In January 1963, suddenly and com- publish his research in major journals. Also Titchmarsh said no, but (encouraged by pletely unexpectedly, he died in his arm- in 1923 he became a Prize Fellow by exam- Ferrar) thought that he might. He sent in chair at home. There were many tributes, ination at Magdalen College, Oxford, an application on a single sheet saying that as everyone was very fond of him. His col- awarded to graduates of outstanding he wished to apply for the geometry Chair league Charles Coulson wrote: ‘There were merit; he held this position for seven years, but could not undertake to lecture on many things about Ted that I have always and was thereby able to keep in touch with geometry as Hardy had done. Two days much admired – his utter humility, which Oxford mathematics. Meanwhile, his later he was appointed and the statutes never betrayed anything but the greatest father had become a church minister in were soon changed to say that the Savilian simplicity; his complete integrity ... and his Essex and Titchmarsh fell in love with Professor of Geometry no longer had to kindness to me when I arrived first; to his Kathleen, the Church Secretary’s daugh- lecture on geometry. Writing to Oswald students (who worshipped him) and to ter. Kathleen called him ‘Oxford’s most B. Veblen about the appointment, Hardy everyone’. But perhaps the final tribute M. [brilliant mathematician]. They mar- said: ‘I fancy Littlewood [one of the elec- ried in 1925 and had three daughters. tors] would have preferred Besicovitch; but In 1929 he was appointed Professor of I expected the electors, with the opportu- Pure Mathematics at Liverpool University, nity of taking a genuinely first rate Oxford succeeding Charles Burkill. While there, product, to do so. The man I am unhappy about is Mordell: first cut out from Cambridge by my decision to stand there, and then here by the Oxford candidate’. The Chair was associated with New College, where he held a number of col- lege positions. Savilian Professor of Geometry During his remaining thirty years in Oxford his research publications contin- ued to appear at a great rate, gradually shifting from number theory and trigono- metrical series to Fourier transforms, eigenfunction expansions, analysis for physicists, and the theoretical background to relativistic quantum mechanics. It was also during this time that he wrote his best- known textbooks. His tract on the E.C. Titchmarsh in later life Riemann zeta-function had been written in Liverpool, but was much developed and can be given in the words of a young visi- rewritten twenty years later in Oxford. His tor to his house who, on being told later first major book, The Theory of Functions that he had been playing dominoes with a was later described as ‘ the rebellion of a great mathematician, remarked: ‘Well, he young, widely read professor against the didn’t seem like it.’ narrow range to which mathematical Reference analysis was then so often confined’. Later Mary Cartwright, Obituary of E. C. Titchmarsh, Journal Titchmarsh receiving an honorary degree from writings included classic texts on Fourier of the London Mathematical Society 39 (1964), 544-565. Sheffield University in 1953 integrals and eigenfunction expansions We thank Jennifer Andrews for supplying these photographs. EMS September 1999 17 SOCIETIES Societies Corner
Societies Corner is a column concerning the was the education of specialist teachers Eidgenössische Technische Hochschule mathematical societies in European countries. in the subject. This is somewhat surpris- (Federal Institute of Technology) or, to The articles in this column could describe the ing in view of the fact that, at the time, give it its German acronym, ETH. history of a particular society or discuss some there were Swiss mathematicians of In keeping with the spirit of the times, event connected with the society. If you feel that international importance. Ch. F. Sturm scientific societies opened their doors in your society would interest others, please contact (1803-55) from Geneva was professor at many university towns. In accordance the column editor, Krzysztof Ciesielski (e-mail: the Ecole Polytechnique and the with the spirit of the enlightenment, [email protected]) in the first instance. Sorbonne in Paris and Jakob Steiner their aim was to bring new knowledge to (1796-1863) from Berne was at the interested parties and circles, by organ- University of Berlin. Mention must also ising gatherings and producing regular Swiss be made of Steiner’s friend, Ludwig publications. The Naturforschende Schläfli (1814-95); although he was Gesellschaft (Scientific Research Society) Mathematical employed as a lecturer at the University of Zurich was founded in 1746, and was Society of Berne, his pay was so low that he one of the first such organisations in could not make a living out of his work. Europe. Its quarterly publication was of Eventually, in 1853, he was made a pro- great importance to the mathematical Urs Stammbach fessor, albeit with a small salary. community; until well into the 20th cen- The Swiss Mathematical Society was Only at the Polytechnicum in Zurich, tury, numerous mathematical essays founded in 1910, fairly late in compari- founded in 1855, was the situation for were published in it by, for example, E. son with similar societies in other mathematicians somewhat better. As a B. Christoffel, R. Dedekind, H. A. European countries. There were, how- basic scientific discipline within the cur- Schwarz, G. F. Frobenius and L. Schläfli, ever, earlier organisations that, at least riculum of a technical education, mathe- and also by H. Weyl and H. Hopf. A partially, served the needs of Swiss math- matics enjoyed a high priority status national Naturforschende Gesellschaft did ematicians. We first look briefly at these right from the beginning. In addition, not come into being until 1815. Within forerunners. in the very early years, and mainly due this body, an informal mathematical sec- to the activity of E. B. Christoffel, a tion was set up and H. A. Schwarz, who Forerunner Organisations school for specialist teachers was estab- was at the ETH, became its first presi- Before 1800 there were few academic lished, where scientific mathematics dent. The activities of the section institutions in Switzerland, but two had could also be pursued within the frame- petered out, however, after Schwarz was already achieved European renown in work of a mathematical seminar. The called to Göttingen in 1875. mathematics by the 18th century. One Polytechnicum quickly made a name for First International Congress of of these was the University of Basle itself as a technical school throughout Mathematicians, 1897 which was founded in 1460 and which, German-speaking Europe, and this The first International Congress of mainly due to the various members of especially applied to the area of mathe- Mathematicians took place in Zurich in the Bernoulli family, became one of the matics. It attracted excellent young 1897. One might be tempted to assume mathematical centres of Europe in the mathematicians from Germany who that this event would have led to the 18th century. The other one was the spent the first years of their scientific founding of a national mathematical Académie de Genève which was founded careers in Zurich: R. Dedekind, E. B. society, but strangely this was not the in 1559. It too achieved European Christoffel, H. A. Schwarz, H. Weber, G. case. Under the supervision of C. F. renown in mathematics, due to people F. Frobenius, A. Hurwitz and H. Geiser, the congress was jointly organ- like G. Cramer (1704-52) and J.-L. Minkowski – all stayed for shorter or ised by mathematicians from the ETH Calandini (1703-58). longer periods. One of the great draw- and the University of Zurich. Other When the other universities in backs for the Polytechnicum in those Swiss universities were not involved in Switzerland were founded in the 19th days was the fact that it did not have the the organisation, nor were many of the century, mathematics did not, apparent- right to confer doctoral degrees. This participants from other Swiss universi- ly, take high priority. All that was became possible in 1909, and in 1911 ties. This was undoubtedly due to the expected of the discipline at that time the name was changed to the
The three founding members of the Swiss Mathematical Society. From left to right, R. Fueter (1880-1950), H. Fehr (1870-1954) and M. Grossmann (1878-1936). 18 EMS September 1999 SOCIETIES ests, while the latter was a talented strengthen the contact and communica- organiser with good connections in tion between mathematicians from the political circles. The so-called ‘Euler different (language) regions. That this Commission’, charged with preparing had been one of the main aims at the the publication of Euler’s complete founding of the society can be seen by works, was founded in the very same the fact that, from the start, the choice of year, in 1907. Under the presidency of the president took the various universi- Rudio, the commission worked swiftly ties of the German and French-speaking and had soon clarified the financial and regions into consideration, taking care scientific aspects of the project. that neither region should dominate the Following the commission’s positive other; the first president, R. Fueter, was report, the project of bringing out at the University of Basle, the second, H. Euler’s complete works in the original Fehr at the University of Geneva and the was started. The first volume appeared third, M. Grossmann at the ETH Zurich. in 1911, and today more than seventy It was, incidentally, mainly due to the volumes have been published. It is initiative of these three men that the expected that the project will be com- society came to be founded. Having pleted within the next few years. mathematics embedded in a truly national society became especially Founding of the Swiss Mathematical important during the first World War as Society the belligerent actions between During the preparations for the pub- Germany and France threatened to poi- lication of Euler’s works, the people son the atmosphere between the French- involved had to fall back on improvisa- speaking Swiss and their German-speak- tion time and again and sorely felt the ing counterparts. need of a national mathematics body in Switzerland: a mathematical organisa- The frontispiece of the Proceedings of the First tion was needed to coordinate and Commentarii Mathematici Helvetici International Congress of Mathematicians, realise an undertaking of these dimen- Scientific demands on mathematics Zurich 1897; the mathematicians featured are sions and to secure funding. This is why continued to grow, and with these grow- (top) Daniel, Jakob and Johann Bernoulli; the publication of Euler’s works led indi- ing expectations the lack of a Swiss peri- (bottom) Leonhard Euler and Jakob Steiner rectly to the founding of the Swiss odical for the publication of scientific Mathematical Society. On 4 September mathematical works became painfully splintered situation and lack of cohesion 1910 the SMG/SMS (Schweizer apparent. A plan was devised during the in the area of mathematics in Mathematische Gesellschaft, its German 1920s to establish such a journal. The Switzerland at the time. name, and Société Mathématique Suisse, its preparation was carried out by H. Fehr, Euler’s Complete Works French one) was founded as a section of together with some of the former presi- During the congress, F. Rudio brought the Schweizerische Naturforschende dents of the society. Faced with the up the idea of publishing the complete Gesellschaft (today, the Swiss Academy of ‘multi-lingualarity’ of Switzerland, it was works of Leonard Euler. The decision to Natural Science). “The advancement of clear from the beginning that neither a do so led indirectly – a few years later – pure mathematics should stand in the French or German title would serve; for to the founding of the Swiss foreground and its promotion within a this reason a Latin name was decided on, Mathematical Society. It was clear to national and international framework” is and Commentarii Mathematici Helvetici was everyone involved in this project that the what Rudolf Fueter, a founding member chosen from three suggestions. publication of Euler’s complete works and the first president of the society, The journal was formally founded at would take many years and require con- wrote in an article in the Neue Zürcher the SMG meeting of 20 May 1928, and siderable sums of money, and it was Zeitung on 26 June 1960, on the occasion R. Fueter was elected as its first editor- clearly necessary that the project be sup- of the 50th anniversary of the SMG. in-chief. In those early years, practically ported by a wide scientific community. The founding of the society in 1910 only articles from mathematicians work- At the celebrations for the 200th was greeted with great interest by Swiss ing in Switzerland were published, in anniversary of Euler’s birth in Basle, F. mathematicians: in its first year the soci- order, as Fueter said in the above-men- Rudio and C. F. Geiser, amongst others, ety numbered 100 members. The annu- tioned NZZ article, “to give as complete submitted a corresponding proposal to al meetings, which took place jointly a picture as possible of what our country Schweizerische Naturforschende Gesellschaft. with the Schweizerische Naturforschende has to offer in the way of mathematics”. Both Rudio and Geiser were at the ETH; Gesellschaft at different venues in In the following years the contributions the former was director of the library Switzerland, were very well attended and published in the Commentarii Mathematici and had wide-ranging historical inter- undoubtedly helped to create and Helvetici became more and more inter- national. Even though the journal filled a void and sold well from the beginning - it had 140 subscribers in its first year of publication - it could not be financed by sales of subscriptions alone. Further funds were needed. In 1929 the SMG therefore decided to create a Stiftung zur Förderung der mathematischen Wissenschaften in order to attract money from the private sector. Eventually, this foundation was also entrusted with other tasks, and today it still supports specifi- The 1994 International Congress logo and a Swiss postage stamp issued to commemorate the cally targeted projects in teaching and Congress. The stamp features a portrait of Jakob Bernoulli, painted by his brother Nicholas, research. together with his law of large numbers. EMS September 1999 19 SOCIETIES International Congress of ship with the European Mathematical Friday 2 February 1883, following the cir- Mathematicians, 1932 Society has included a prominent posi- culation of a letter from A. J. G. Barclay, A. In 1932 the ICM took place in Zurich for tion: the current president, Rolf Jeltsch Y. Fraser and C. G. Knott to ‘Gentlemen in the second time. R. Fueter from the from the ETH, is also a member of the Edinburgh, in Cambridge, and throughout University of Zurich and M. Plancherel SMG board. Scotland generally whom they deemed from the ETH had received a request in likely to take an interest in such a Society.’ Bologna that the next ICM take place in Elemente der Mathematik The letter proposed the establishment, in In 1946 L. Locher-Ernst started publica- Zurich. The preparations were very dif- connection with the University, of a society tion of the journal Elemente der ferent from those for the 1897 congress. for the mutual improvement of its mem- Mathematik, quite independent of the Care was taken to carry them out under bers in the mathematical sciences, pure SMG. It was aimed at a readership con- the patronage of the SMG and that the and applied. Methods suggested for the sisting mostly of teachers of higher edu- organising committee should include attainment of this object were reviews of cation, although it dealt primarily with people from universities from other work, both British and foreign, historical mathematics as science and not didac- parts of Switzerland. In a far greater notes, discussion of new problems or new tics. In this, Elemente der Mathematik was measure than in 1897, this congress was solutions, and comparisons of the various following aims similar to those of organised with the mutual participation systems of teaching in different countries. L’Enseignement Mathématique, the journal of the entire Swiss mathematical com- It is interesting that schoolteachers took set up in 1899 by H. Fehr and Ch. munity. R. Fueter from the University of a leading part in the foundation of the Laisant at Geneva. In 1975 ownership of Zurich served as president and there Society. Mr Barclay and Mr Fraser were Elemente der Mathematik was transferred were two vice-presidents, M. Plancherel teachers at George Watson’s College in to the SMG and the Society has been from the ETH and H. Fehr from the Edinburgh, while Cargill Knott, who took charged with its publication ever since. University of Geneva. the chair at the inaugural meeting, was International Congress of assistant to the professor of natural philos- The Steiner and Schläfli Archives Mathematicians, 1994 ophy in the University of Edinburgh, Peter As early as 1930 the SMG had laid the Searching for a venue for the 1994 Guthrie Tait. At the first meeting, foundation for a Steiner archive and International Congress of Professor Tait and George Chrystal, the formed a responsible committee. The Mathematicians, the IMU again turned professor of mathematics at Edinburgh, archive was to preserve and manage J. to the SMG with the request that it take were elected honorary members. Steiner’s scientific legacy; his Complete place in Switzerland. After a short loca- Professor Chrystal gave an address on Works had been published by Weierstrass tion assessment, Zurich was chosen, for ‘Present fields of mathematical research’, in 1881-82. In 1937 the task of the com- the third time, as the most suitable and 51 people joined as ordinary mem- mittee was extended to include “sifting place. Once again, as in 1932, a com- bers. J. S. Mackay, chief mathematics mas- through the legacies of the two great mittee was put together under the ter at the Edinburgh Academy, was elected Swiss mathematicians Steiner and patronage of the SMG, made up of as first president of the Society, with Dr Schläfli and making their work accessi- mathematicians from all over Knott as both secretary and treasurer. Mr ble”. Schläfli’s work was eventually pub- Switzerland. H. Carnal from the Fraser succeeded Dr Knott in these offices lished in 1956 in three volumes. University of Berne served as president from the autumn of 1883, when the latter International Contacts and Chr. Blatter from the ETH Zurich left to become professor of physics at the The meetings of the SMG brought many was secretary. This congress provided a Imperial University of Japan; Cargill Knott opportunities to invite renowned foreign welcome opportunity to steer the inter- was an authority on magnetism and seis- mathematicians to Switzerland to give ests of the general public towards math- mology, and was responsible for conduct- lectures. For the smaller Swiss universi- ematics and elucidate its role in today’s ing the magnetic survey of Japan. After his ties, these invitations provided among technological society. At the opening return to a lectureship in mathematics at the few opportunities – until well into address, the Federal Minister of Home Edinburgh in 1891, he served twice as the the 1940s – for making international Affairs, Ruth Dreifuss, picked up on this Society’s president, in 1893-94 and in contacts. Furthering international rela- very point: ‘It is the task of the scientific 1918-19. His name is known to many tions thus remained an especially impor- community to tell the public why science thousands, including the present writer, tant task of the SMG. With this in mind, matters. It is your task and mine.’ The the society created numerous honorary role of our scientific society towards soci- memberships for foreign mathemati- ety as a whole cannot be better cians with whom it wished to foster rela- expressed. tions. These included, amongst others, R. Dedekind, D. Hilbert, H. Weyl, R. Urs Stammbach is Professor of Mathematics Nevanlinna, G. Pólya, H. Whitney and J. at the ETH Zurich. He wishes to thank Dr. Leray, as well as worthy Swiss mathe- F. Lanini from the Wissenschaftshistorische maticians (we name only deceased col- Sammlungen der ETH Bibliothek for her leagues) R. Fueter, C. F. Geiser, H. critical proof reading of this manuscript and Hopf, A. M. Ostrowski, A. Pfluger, M. for many useful suggestions. Plancherel, G. de Rham, W. Saxer and B. L. van der Waerden. The ever stronger relationship Edinburgh between Switzerland and the International Mathematical Union must Mathematical be mentioned in this context. H. Hopf was president from 1955-58, while B. Society Eckmann served as secretary. Later presidents of the IMU from Swiss uni- Philip Heywood versities were G. de Rham from the uni- versities of Geneva and Lausanne, and The Edinburgh Mathematical Society was Professor Ian Sneddon of Glasgow University K. Chandrasekharan and J. Moser from formed at a meeting in the mathematics and Professor Emeritus W. L. Edge of the ETH Zurich. Recently, the relation- classroom of Edinburgh University on Edinburgh University enjoy a joke at the Society’s centenary dinner in February 1983. 20 EMS September 1999 SOCIETIES Department of Mathematics and Statistics, James Clerk Maxwell Building, Mayfield Road, Edinburgh EH9 3JZ, Scotland, or on the internet at http://www.maths.ed.ac.uk /~chris/ems. The Society has reciprocity agreements with a number of overseas mathematical societies, through which membership of one society enables one to join others at a reduced subscription. Postgraduate students at Scottish universi- ties are offered free membership of the Society for up to three years to encourage them to take part in the Society’s activities. The Society’s library is housed in the James Clerk Maxwell Building at the University of Edinburgh, and consists mainly of periodicals obtained by exchange with other learned societies throughout the world. Members may con- sult or borrow books from the Society’s library, and are also entitled to use the Edinburgh University Library. George Chrystal and Peter Guthrie Tait, the first honorary members of the Society. Each year since 1983, the Society has set who were school pupils before the age of Meetings are normally on Friday after- aside a sum of money, known as the calculators, through the publication of his noons, and are open to all who are inter- Centenary Fund, to give financial support booklet of four-figure mathematical tables. ested. Speakers are chosen to cover a wide to a variety of mathematical activities, Prominent and faithful early members range of topics in pure and applied math- including research visits, conferences and of the Society included George Alexander ematics. From time to time, the Society publications. Any member of the Society Gibson, president of the Society in 1888- organises joint meetings with the London may apply for a grant or guarantee from 89, professor of mathematics at the Mathematical Society, and regularly hosts this fund. Awards are normally paid to an Glasgow and West of Scotland Technical in Scotland that Society’s Popular applicant’s institution rather than to the College from 1895-1909 and at Glasgow Lectures, intended for senior school pupils applicant personally, and a key feature in University from 1909-27, and Robert and other non-professional mathemati- the consideration of an application is the Franklin Muirhead, president in 1899- cians. perceived benefit to the wider mathemati- 1900. Each year the Society holds a meeting of cal community. Part of the Centenary With the growth in size of university postgraduate students at the Burn at Fund each year is earmarked for applica- departments, the Society gradually came Edzell, and every four years it holds the St tions from members resident overseas. more under their influence and paid more Andrews Colloquium, a major conference The Society also sets aside each year a sum attention to mathematical research. Sir held for a week in July, at which three or of money known as the Education Fund, to Edmund Whittaker, professor of mathe- four distinguished international speakers give financial support to educational activ- matics at the University of Edinburgh from give short courses of lectures; the next such ities of a mathematical nature. As with the 1912-46, was responsible for the first colloquium will be in 2001. Centenary Fund, applications may be mathematical colloquium sponsored by the The Proceedings of the Edinburgh made by members of the Society, but addi- Society, in 1913. A second colloquium was Mathematical Society have been published tionally the Education Committee itself held in Edinburgh in 1914, before the out- by the Society since 1884. Three issues are actively seeks out projects worthy of sup- break of war. After the war, colloquia were published each year, containing research port. In addition to the Centenary and resumed at St Andrews in 1926, largely papers covering a wide range of topics in Education Fund schemes, the General through the enthusiasm of H. W. Turnbull, pure and applied mathematics. The Committee of the Society from time to who succeeded to the Regius Chair of Proceedings has an international board of time makes special grants. Sometimes mathematics there in 1921, and colloquia consulting editors and has a world-wide these are major grants for the support of have been held in St Andrews regularly circulation. large international conferences, but equal- ever since. The Society is now firmly estab- The Society is conscious that its ly they may be small grants for worthy lished as the principal mathematical soci- declared objective, “the promotion and mathematical purposes that do not fall ety for the university community in extension of the Mathematical Sciences”, is within the remits of the two schemes. Scotland. Its membership, which exceeds sometimes best achieved through the sup- 400, is drawn from the Scottish universities port of activities planned and instigated by and other educational institutions, as well others. It therefore has schemes for Acknowledgement I have drawn freely on the as from mathematicians in industry and awarding grants for mathematical activi- history of the Society, “The First Hundred commerce both at home and overseas. ties, and it joins with other Societies in sup- Years 1883-1983”, Proc. Edinburgh Math. The first meeting outside Edinburgh porting various initiatives at national and Soc. 26 (1983), 135-150, by Robert Rankin, was held in Glasgow in March 1900 during international level through subscriptions, formerly professor of mathematics at Muirhead’s Presidency. Regular meetings donations and the sending of representa- Glasgow University, President of the in Glasgow followed, and the Society began tives. Society from 1957-58, and currently an to meet annually in St Andrews in 1922. Membership of the Society is open to all Honorary Member, and also on the The first meetings in Dundee and who are interested. Prospective members Annual Report for 1997-98 by the present Aberdeen were held in 1930 and 1937, are nominated at one of the Society’s meet- Honorary Treasurer, Neil Dickson of respectively. At present, the Society meets ings and elected at the following meeting. Glasgow University. regularly in ten different universities. Currently membership costs 11 pounds Eight ordinary meetings are held in each sterling per year, or 20 pounds including Session, from October to June, three at the an individual subscription to the Philip Heywood teaches in the Department of University of Edinburgh, and the remain- Proceedings. Further information is avail- Mathematics and Statistics at the University of der in other Scottish universities. able from the Honorary Secretary, Edinburgh. EMS September 1999 21 CONFERENCES Programme committee: Christophe Croux, Alois Kneip, Lucrezia Reichlin, Eric Renault, Bas Werker FForthcomingorthcoming conferconferencesences Organisers: Institut de Statistique et de Recherche Opérationelle, European Center for Advanced Research in compiled by Economics and Statistics Kathleen A S Quinn Site: Universite Libre de Bruxelles, Brussels Please e-mail announcements of European con- 21-26: Foundation Workshop on Deadline: for submission of papers, ferences, workshops and mathematical meetings Stochastics and Quantum Physics, already passed of interest to EMS members, to Aarhus, Denmark Information: contact RFBS99, Institut de [email protected]. Announcements Aim: to foster fruitful discussions and col- Statistique, CP210, ULB, B-1050 should be written in a style similar to those laboration on the role and use of stochas- Bruxelles, Belgium; tel/fax: +32-2-650- below, and sent as text files (but not as TeX tics in quantum physics, by bringing 5899 input files). Space permitting, each announce- together leading physicists and mathe- e-mail: [email protected] ment will appear in detail in the next issue of the maticians having an active interest in the URL: http://isro.ulb.ac.be Newsletter to go to press, and thereafter will be themes of the workshop December 1999 briefly noted in each new issue until the meeting Main themes: (confirmed key-note speak- takes place, with a reference to the issue in which ers in parentheses) laser physics/quantum 2-4: Conference on Mathematical and the detailed announcement appeared. The pre- optics (F. Bardou, H. Carmichael, G. Computational Methods in Music, sent issue includes conferences up to September Mahler), quantum stochastic processes (A. Vienna 2000. Barchielli, V.P. Belavkin, G. Lindblad), [Part of the EMS Diderot Forum series] Wick products, white noise analysis and October 1999 Aim: to bring together mathematicians Malliavin calculus (B. Øksendal), the role interested in or working on subjects of of generalised measurements and quan- relevance to music, and musicians who 4-8: International Workshop on General tum statistical inference (R. Gill, S. Topological Algebras, Tartu, Estonia make use of computational methods; to Massar, H. Wiseman), quantum informa- raise public awareness of the role of math- Information: tion (A.S. Holevo, P. Høyer, A. Peres, S. e-mail: [email protected] ematics in our society, in particular by Popescu) showing the multiple connections and 4-8: ParaOpt VI, 6th International Programme: during the first two days, B. mutual influences which exist between Conference on Parametric Optimization Øksendal will give a short course on Wick music and mathematics and Related Topics, Dubrovnik, Croatia products, Malliavin calculus and their Topics: synthesis of musical sounds, Information: applications in physics analysis of musical sounds (transcription URL: Organisers: O.E. Barndorff-Nielsen digital data - MIDI, time-frequency meth- http://www.math.hr/dubrovnik/index.htm (Aarhus), K. Molmer (Aarhus) ods, quantitative analysis of instruments Fee: the registration fee is DKK600, to 4-15: Isaac Newton Institute Workshop, and musical interpretation), restoration cover lunches, coffee during breaks and and improvement of old recordings and Defect Mechanics and Non-locality, the conference dinner. Those who partici- Cambridge, UK mathematical algorithms, ‘instrument pate only in the short course need not pay optimisation’ (instrumental parameter Information: the registration fee URL: http://www.newton.cam.ac.uk/programs/ identification), coding strategies for musi- Site: University of Aarhus cal information, mathematical models for 5-9: Géométrie des équations differen- Deadlines: for registration, already passed musical sound or rhythm tielles, Luminy, France Information: Invited speakers: (confirmed) W. Information: contact J.-P. Francois, URL: Fitzgerald (Cambridge), G. DePoli Equipe «Géométrie différentielle, http://www.maphysto.dk/events/QuantumStoc (Padua), X. Serra (Barcelona), G. Systèmes dynamiques, applications, UFR 99/index.html Wakefield (Ann Arbor, Michigan) 920, Mathématiques, B.C. 172, Tour 46, November 1999 Scientific committee: Werner A. Deutsch 5ème étage, Université P.-M. Curie, Paris (Austrian Acadademy of Science, VI 75252, Paris, France; fax: +33-1-442- Kommission für Schallforschung), Hans 5345 1-12: Isaac Newton Institute Workshop, Models of Fracture, Cambridge, UK G. Feichtinger (Institut für Mathematik, e-mail: [email protected] University of Vienna, main organizer), [For details, see EMS Newsletter 32] Information: URL: http://www.newton.cam.ac.uk/programs/ Christian Krattenthaler (Institut für 6-9: New Trends in the Calculus of Mathematik, University of Vienna), Erich Variations, Lisbon, Portugal 2-5: Workshop on Hilbert’s 10th prob- Neuwirth (Institut für Statistik, University Information: Contact CMAF/Univ. lem, relations to arithmetic and algebra- of Vienna), Gregor Widholm (Institut für Lisboa, Av. Prof. Gama Pinto 2, 1649-003 ic geometry Wiener Klangstil, University of Music and Lisboa, Portugal Information: contact Jan Van Geel or Dramatic Arts, Vienna), Gerhard Widmer e-mail: [email protected] Karim Zahidi, University of Gent, (Austrian Research Institute for Artificial URL: http://www.math.ist.utl.pt/~ntcv99/ Department of Pure Mathematics, Intelligence, Vienna) [For details, see EMS Newsletter 32] Galglaan 2, B-9000 Gent, Belgium; fax. Local organising committee: Monika +32-9-264-49 93 Dörfler, Hans G. Feichtinger, Christian 15-16: Two-day LMS meeting, New e-mail: [email protected] Krattenthaler, Stefan Thurner Applications of Twistor Theory, London, URL: Supporting institutions: Universität UK http://cage.rug.ac.be/~hilbrt10/hilbert10.html Wien, EMS (European Mathematical Speakers: K.P. Tod (Oxford), M.G. [For details, see EMS Newsletter 32] Society), OEMG (Austrian Mathematical Eastwood (Adelaide), N.J. Hitchin 25-26: XXème Rencontre Franco-Belge Society), Mathematischer Zirkel der (Oxford), S. Merkulov (Glasgow), L.J. Universität Wien, OECG (Austrian Mason (Oxford), Sir Roger Penrose de Statisticiens, Brussels, Belgium Topic: factor models. In a factor model, Computer Society), Bank Austria (Oxford) Proceedings: to be published in a special Grants: apply to [email protected] one tries to explain the correlation between several random variables using a issue of the OECG series (Austrian Deadline: to book reserved accommoda- Computer Society) tion, 1 October 1999 limited number of factors. This confer- ence aims to present the latest develop- Site: Kleiner Festsaal der Universität Information: Wien, main building of the University of e-mail: [email protected] ments as well as the numerous applica- tions of factor models Vienna 22 EMS September 1999 CONFERENCES Accommodation: participants should Marcati, D. Marchesin, K.W. Morton, B. Harmonic maps and curvature properties make their own arrangements; the organ- Perthame, D. Serre, E. Tadmor, A. Tveito, of submanifolds, Leeds; 14-17 April, isers may be consulted and G. Warnecke. Mathematical methods of regular dynam- Social event: a concert with contributions Organisers: G. Warnecke, H. ics - dedicated to the 150th anniversary of from participants is planned Freistueühler. Sonja Kowalevski, Leeds; 16-17 April, Deadline: for submission, already passed Call for papers: abstracts for contributed British topology meeting, Sheffield; 17 Information: contact Hans G. papers may be submitted immediately April, Model theory, Leeds Feichtinger, Institut für Mathematik, Site: Otto-von-Guericke University, Information: Universität Wien, Strudlhofgasse 4, A- Magdeburg e-mail: [email protected] 1090 Vienna Information: contact HYP-2000 c/o URL: http://www.amsta.leeds.ac.uk/bmc/ tel: +43-1-4277-50696; fax: +43-1-4277- Institut für Analysis und Numerik Otto- 26-28: Mathematical Education of 50620 von-Guericke-Universität Magdeburg PSF Engineers, Loughborough, UK e-mail: [email protected] 4120, D-39016 Magdeburg, Germany; Aim: to reflect the progress and experi- URL: http://tyche.mat.univie.ac.at/~diderot/ fax: HYP-2000 at +49-391-67-18073 ences of initiatives within the teaching of 20-22: Seventh IMA International e-mail: [email protected] mathematics to engineers in recent years, Conference on Cryptography and magdeburg.de to debate areas of known concern and to Coding, Cirencester, UK URL: http://rubens.math.uni-magdeburg.de/ learn together from current best practice; Information: ~hyp2000 to examine collectively as mathematicians, e-mail: [email protected] March 2000 academic engineers and engineers in URL: http://www.ima.org.uk/mathematics/ industry the engineering mathematics conferences.htm 11-12: School Mathematics 2000, provision for the future [For details, see EMS Newsletter 32] Helsinki, Finland Programme: invited speakers, contributed papers (or posters), workshop sessions January 2000 Information: e-mail: [email protected] and a forum Organising committee: Leslie Mustoe 17-22: Workshop on Computational April 2000 (Loughborough), Stephen Hibberd Stochastics, Aarhus (Nottingham), Trevor Easingwood (IMA), Information: contact Eva B. Vedel 11-14: Workshop on Harmonic Maps Duncan Lawson (Coventry); Heather Jensen, Department of Mathematical and Curvature Properties of Liddell (London), John McWhirter Sciences, University of Aarhus, Ny Submanifolds 2, Leeds, UK (DERA), Stephen Reid (UMIST), Ralph Munkegade, DK-8000 Aarhus C Aim: to stimulate cooperation between Smith (Jaguar) e-mail: [email protected] researchers working in harmonic maps Proceedings: to be published by the URL:http://www.maphysto.dk/events/Comp and those working in submanifold theory Institute of Mathematics Stoc2000/ Speakers: (expected plenary speakers) F. Sponsors: The Institution of Civil [For details, see EMS Newsletter 32] Burstall, T.E. Cecil, M.A. Guest, F. Engineers and The Institution of Chemical Engineers February 2000 Helein, Y. Ohnita, F. Pedit, U. Pinkall, G. Thorbergsson Site: Loughborough University Programme committee: S. Carter, J.C. Deadline: for abstracts, 3 December 3-5: Mathematics Today, Trondheim, Information: Norway Wood. Information: contact J.C. Wood, School URL: http://www.ima.org.uk/mathematics/ Note: primarily intended for a conferences.htm Scandinavian audience of Mathematics, University of Leeds, Leeds LS2 9JT, UK Information: May 2000 URL: http://www.math.ntnu.no/talltiltusen/ e-mail: [email protected] URL: http://www.amsta.leeds.ac.uk/pure/ 28-3 March: Eighth International geometry/leeds2000.html 29-9 June: Foliations: Geometry and Conference on Hyperbolic Problems, Dynamics Revisited, Banach Centre, Magdeburg, Germany 17-20: 52nd British Mathematical Warsaw, Poland Topic: theory, numerics and applications Colloquium, Leeds, UK Aim: the exchange of scientific informa- of hyperbolic conservation laws and relat- Scope: the annual meeting of (pure) tion among specialists in the theory of ed fields mathematicians in the UK; all mathemati- foliation and related topics, in particular Plenary speakers: Yann Brenier (Paris), cians are welcome in the area of relations between this theo- Thomas Hou (Pasadena), Shuichi Plenary speakers: Sir Michael Atiyah ry, differential geometry, dynamical sys- Kawashima (Fukuoka), Ingo Mueller (Edinburgh), Simon Donaldson (Imperial tems and ergodic theory (Berlin), Alfio Quarteroni (Lausanne), College), Vaughan Jones (Berkeley), Phil Roe (Ann Arbor), Giovanni Russo Harvey Friedman (Ohio), Jens Jantzen June 2000 (L’aquila), Steve Schochet (Tel Aviv), Joel (Aarhus); also 14 main speakers, mostly Smoller (Ann Arbor), Michael Struwe based in the UK 13-16: First AMS-Scandinavian (Zuerich), Kevin Zumbrun (Bloomington) Special sessions: harmonic maps and International Mathematics Meeting, Invited Speakers: Francois Bouchut minimal surfaces; operator algebras XXIII Scandinavian Congress of (Orleans), Suncica Canic (Houston), Splinter groups: number theory, mathe- Mathematicians, Odense, Denmark Pierre Degond (Toulouse), Eduard matical logic, algebraic topology, mathe- Speakers: Tobias Colding (New York), Feireisl (Prague), Emmanuel Grenier matical education, functional analysis, Nigel J. Hitchin (Oxford), Elliott Lieb (Lyon), David Hoff (Bloomington), Shi Jin algebra, and integrable systems (Princeton) Pertti Mattila (Jyväskylä), (Atlanta), Kenneth H. Karlsen (Bergen), Organisers: H.G. Dales and H.D. Curtis McMullen (Harvard), Alexei Smadar Karni (New York), Claus-Dieter Macpherson, Leeds Rudakov (Trondheim), Dan-Virgil Munz (Stuttgart), Benedetto Piccoli Site: University of Leeds Voiculescu (Berkeley), Johan Hostad (Salerno), Bradley Plohr (Stony Brook), Registration: forms will be available in (Stockholm) Ed Seidel (Potsdam), Tao Tang (Hong early 2000; they will be circulated with the Special sessions: (to run in parallel) alge- Kong), Eleuterio Toro (Manchester), London Mathematical Society Newsletter, braic groups/representation theory, com- Cheng-Chin Wu (Los Angeles), Tong and sent to all UK Departments of plex analysis in higher dimensions, differ- Yang (Hong Kong), Shi-Hsien Yu (Osaka) Mathematics; to obtain a form, please ential geometry, discrete mathematics, Scientific committee: J. Ballmann, A. contact H.D. Macpherson at the e-mail dynamical systems, geometric Bressan, C. Dafermos, B. Engquist, M. address below analysis/PDE, K-theory and operator alge- Feistauer, H. Freistühler, J. Glimm, L. Accommodation: available on the campus bras, linear spaces of holomorphic func- Hsiao, R. Jeltsch, P. Lax, T.-P. Liu, P. of the University of Leeds tions, mathematical physics, mathematics Satellite conferences: 11-14 April, education, stochastic DE and financial EMS September 1999 23 CONFERENCES mathematics, joint EWM and AWM ses- (Paris), F. Hirzebruch (Bonn), K.-H. 33095 Paderborn, Germany or sion Hoffmann (Bonn), S. Müller (Leipzig). Universidad Politècnica de Valencia, Information: contact Hans J. Munkholm, Local organiser: K. Hulek Departamento de Matemática Aplicada, Odense University, Campusvej 55, DK Note: the number of junior participants is E-46071 Valencia, Spain 5230 Odense M, Denmark limited to 50. The workshop is aimed at e-mail: [email protected] tel: +45-6557-2309/+45-6593-2691 younger mathematicians, that is, those URL: http://math-www.uni-paderborn.de e-mail: [email protected] who have had a PhD for at least one year /VLC2000 URL:http://www.imada.ou.dk/~hjm/AMS. and have done research beyond their the- [For details, see EMS Newsletter 32] Scand.2000.html sis, but do not yet hold a senior tenured 3-7: Sixth International Conference on 14-17: International Workshop for position p-Adic Analysis, Ioannina, Greece Operator Theory and Applications Deadline: for applications, 31 January Information: (IWOTA), Bordeaux, France Information: contact K. Hulek, Institut für e-mail: [email protected] Mathematik, Universität Hannover, Aim: to bring together mathematicians 4-7: Second International Conference on and engineers interested in operator the- Postfach 6009, D-30060 Hannover, Germany Mathematical Methods in Reliability, ory and its applications to related fields Bordeaux, France Scope: operator theory and related topics e-mail: [email protected] URL: http://www-ifm.math.uni-hannover.de Information: in mathematics and its applications: dif- e-mail: [email protected] ferential operators, reproducing kernel /info/perspectives.html spaces, harmonic analysis, control theory, 26-30: POISSON 2000, France 10-14: IUTAM Symposium on Free system theory and signal processing Information: Surface Flows, Birmingham, UK Invited speakers: (confirmed) V.M. e-mail: Topics: axisymmetric free surface flows, Adamyan, D. Alpay, A. Böttcher, L de [email protected] moving contact lines, non-linear water Branges, R.F. Curtain, K. Davidson, C. waves, collapsing bubbles 28-1 July: First World Congress of the Scientific committee: J.R. Blake, J.B. Foias, I. Gohberg, W. Helton, R. Bachelier Finance Society, Paris, France Kaashoek, A. Megretski, Y. Meyer, S.N. Keller, A.C. King, W. Lauterborn, D.H. Information: Peregrine, A. Prosperetti, E.O. Tuck, L. Naboko, J. Partington, G. Pisier, C. e-mail: [email protected] Sadosky, E.B. Saff, K. Seip, O. Staffans, S. van Wijngaarden Treil, S. Verduyn Lunel 29-3 July: International Workshop on Local organising committee: J.R. Blake Organising committee: L. Baratchart Nonlinear Spectral Theory, Würzburg, (co-chair), A.C. King (co-chair), J. (Sophia-Antipolis), A. Borichev Germany Billingham, S.P. Decent, Y.D. (Bordeaux), G. Cassier (Lyon), J. Esterle Topic: state-of-the-art of spectral and Shikhmurzaev, J.R. King, J.R.A. Pearson, (Bordeaux), N. Nikolski, chairman eigenvalue theory for nonlinear operators E.J. Hinch, J.M. Vanden-Broeck (Bordeaux and St.Petersburg), V.-H. Scope: to bring together experts on non- Site: University of Birmingham Vasilescu (Lille) linear analysis to discuss recent develop- Related meetings: 17-20 July, IUTAM President of the steering committee: I. ments and open problems in the theory, Symposium 2000/10 Diffraction and Gohberg (Tel Aviv) methods, and applications of spectra of Scattering in Fluid Mechanics and Site: University of Bordeaux-1 nonlinear operators Elasticity, Manchester; 17 July-4 August, Information: Programme: 15 invited one-hour lectures, Free Boundary Problems in Industry, e-mail: [email protected] short communications, informal discus- Isaac Newton Institute, Cambridge, UK URL: http://www.math.u-bordeaux.fr/~iwota/ sion Deadline: for abstracts, 1 January Lecturers: R. Chiappinelli (Siena), G. Information: 18-21: International Conference on Conti (Florence), E. De Pascale (Cosenza), URL: Monte Carlo Simulation, Monte Carlo, W. Feng (Toronto), M. Furi (Florence), M. http://www.mat.bham.ac.uk/research/iutam.htm Monaco Martelli (Fullerton), M.Z. Nashed Aim: to provide an opportunity for engi- 10-14: Third European Congress of (Newark), T. Riedrich (Dresden), P. Mathematics, Barcelona neers, mathematicians and other profes- Santucci (Rome), C.A. Stuart (Lausanne), sionals who are interested in the theoreti- Information: contact Societat Catalana de V.A. Trenogin (Moscow), M. Vath Matemátiques, Carrer del Carme, 47, E- cal and practical aspects of Monte Carlo (Wurzburg), A. Vignoli (Rome), J.R.L. simulation to exchange ideas on the status 08001 Barcelona Webb (Glasgow), P.P. Zabrejko (Minsk) tel: +343-270-16-26; fax: +343-270-11- of MCS procedures Sponsor: Deutsche Topics: include algorithms for random 80 Forschungsgemeinschaft (DFG), Bonn e-mail: [email protected] number generation, methods for solutions Site: Department of Mathematics, (spectral simulation, solving the PDEs URL: http://www.iec.es/3ecm/ University of Würzburg (Franconia) [For details, including satellite confer- directly, numerics of PDEs), algorithms Information: contact Jurgen Appell, (evolutionary, genetic), practical engineer- ences, see First Announcement in EMS Department of Mathematics, University of Newsletter 31] ing applications Würzburg, Am Hubland, D-97074 Deadline: for abstracts, 12 December Würzburg, Germany 17-20: IUTAM Symposium 2000/10 Information: tel: +49-931-8885017; fax: +49-931- Diffraction and Scattering in Fluid URL: 8885599 Mechanics and Elasticity, Manchester, http://www.uibk.ac.at/c/c8/c810/conf/mcs_ e-mail: [email protected] UK 2000.html URL: www.mathematik.uni-wuerzburg.de Aim: to bring together researchers from a 18-24: Perspectives of Mathematics, /~appell/nlst.html range of different subject disciplines and Goslar, Germany industrial focuses, who employ common [World Mathematical Year event] July 2000 techniques and approaches, to examine Aim: to bring together senior and the ways that the various subjects have younger mathematicians to discuss the 3-7: ALHAMBRA 2000, Granada, Spain developed and to stimulate cross-fertilisa- perspectives of mathematics at the turn of Information: tion of the theoretical ideas and method- the century e-mail: [email protected] ologies. Speakers: (confirmed) V. Arnold (Paris 3-7: Functional Analysis Valencia 2000, Themes: elastic waves, acoustic phenome- and Moscow), M. Gromov (Bures sur Spain na in stationary fluids, aeroacoustics, dif- Yvette), R. James (Minnesota), J.P. Morel [satellite conference to the Third fraction of free surface and other geo- (Cachan), G.C. Papanicolaou (Stanford), European Congress of Mathematics in physical waves A. Schrijver (Amsterdam), D. Zagier Barcelona, 10-14 July] Organisers: hosted jointly by the (Bonn) Information: contact: K.D. Bierstedt or J. Departments of Mathematics, University Scientific committee: J.-P. Bourguignon Bonet, Univ. Paderborn, FB 17, Math., D- of Manchester and Keele University Local organising committee: I. David 24 EMS September 1999 CONFERENCES Abrahams (Manchester, chair), C. John foundations of computer science, set theo- Stavrinos (Greece), L. Tamassy (Hungary), Chapman (Keele), Paul A. Martin ry, model theory, computability and com- K. Trencevski (Macedonia), I. Vaisman (Manchester), Mike J. Simon plexity theory, history of 20th-century (Israel), L. Vanhecke (Belgium), L. (Manchester). Graham Wilks (Keele), logic, philosophy, applications of logic to Verstraelen (Belgium), E. Vassiliou Andrew J. Willmott (Keele) cognitive sciences (Greece) Proceedings: will be published Programme: 24 plenary talks, four three- Organiser: C. Udriste Site: University of Manchester hour tutorials, parallel sessions of con- Proceedings: selected papers will be pub- Related conference: IUTAM Symposium tributed talks. lished in the Balkan Journal of Geometry on Free Surface Flows, 10-14 July 2000 Programme committee: Daniel Andler and its Applications Deadline: for abstracts, 31 January (Paris), Chantal Berline (Paris), Barry Site: University Politehnica of Bucharest, Information: contact Professor David Cooper (Leeds), Akihiro Kanamori Splaiul Independentei Street 313, Abrahams, Department of Mathematics, (Boston), Charles Parsons (Harvard), Bucharest, RO-77206 University of Manchester, Oxford Road, Alexander Razborov (Moscow), Helmut Deadline: for registration, 15 April Manchester M13 9PL, UK Schwichtenberg (Munich), John Steel Information: contact V. Balan, University tel: +44-161-275-5901, fax: +44-161-275- (Berkeley), Stevo Todorcevic (Paris), Dirk Politehnica of Bucharest, Department of 5819 van Dalen (Utrecht), Alex Wilkie Mathematics I, Splaiul Independentei e-mail: [email protected] (Oxford), Carol Wood (Chairperson, 313, RO-77206, Bucharest, Romania URL: http://www.keele.ac.uk/depts/ma/iutam/ Wesleyan University) fax: +401-411-53-65 17-22: Colloquium on Lie Theory and Organising committee: Chantal Berline e-mail: [email protected] (Paris), Zoé Chatzidakis (Paris), René Cori Applications, Vigo, Spain August 2000 [Satellite Activity of the Third European (Chairman, Paris), Maximo Dickmann Congress of Mathematics] (Paris), Jacques Dubucs (Paris), Jean- Baptiste Joinet (Paris), Daniel Lascar 8-12: XVIII Nevanlinna Colloquium, Programme: three courses of three hours Helsinki, Finland each, eleven invited lectures and several (Paris), Yves Legrandgérard (Paris), Jean Mosconi (Paris), Marie-Hélène Mourgues [World Mathematical Year event] short communications Scope: the emphasis will be on subjects in Courses: D.V. Alekseevsky, Semisimple (IUFM de Créteil), Catherine Muhlrad- Greif (Paris), Leszek Pacholski (Wroclaw), some way connected to analysis, especially Lie algebras, Dynkin diagramms and geometric aspects geometry of flag manifolds; A.T. Jean-Pierre Ressayre (Paris), Boban Velickovic (Paris), Francoise Ville (Paris) Speakers: include A. Baernstein (St. Fomenko, Lie groups and integrable Louis), J. Cheeger (New York), W. Hamiltonian systems; M. Scheunert Site: the Sorbonne (Université Paris 1) Deadline: for abstracts, 31 March Bergweiler (Kiel), P. Mattila (Jyväskylä), Speakers: S. Benayadi, M. Bordemann, V. C. Bishop (Stony Brook), C. McMullen Cortés, A. González-López, Yu.B. Information: to receive the congress announcements, send a request by e-mail (Berkeley), B. Bowditch (Southampton), Hakimjanov, E. Koelink, M. de León, E. Yu. Reshetnyak (Novosibirsk), L. Carleson Macías-Virgós, A. Medina, C. Moreno, K- to [email protected] with subject get-announcement, or by fax to (Stockholm) H. Neeb Programme: about 45 invited talks Scientific committee: D.V. Alekseevsky +33-1-44-27-61-48 (from abroad) and 01- 44-27-61-48 (from France), or by letter to Organising committee: Peter Buser (Germany), S. Benayadi (France), M. (Lausanne), Seppo Rickman (Helsinki), Bordemann (Freiburg), V. Cortés (Bonn), LC2000, UFR de Mathématiques, case 7012, Université Paris 7-Denis Diderot, 2 Ilpo Laine (Joensuu), Kurt Strebel A.T. Fomenko (Moscow), A. González- (Zürich), Olli Lehto (Helsinki),.Pekka López (Madrid), Yu.B. Hakimjanov place Jussieu, 75251 Paris Cédex 05, France. Tukia (Helsinki), Olli Martio (Helsinki), (France), K.H. Hofmann (USA), E. Matti Vuorinen (Helsinki) Koelink (Netherlands), M. de León e-mail: [email protected] URL: http://lc2000.logique.jussieu.fr Site: University of Helsinki (Spain), E. Macías-Virgós (Santiago), A. Information: to receive the second Medina (France), C. Moreno (Bourgogne 31-3 August: Third Conference of announcement, send e-mail to and Madrid), K-H. Neeb (Germany), M. Balkan Society of Geometers, Bucharest, [email protected] or write Scheunert (Bonn) Romania to Riitta Ulmanen, Department of Organising committee: N. Alonso, I. Topics: Riemannian geometry, symplectic Mathematics, P.O. Box 4 (Yliopistonkatu Bajo, R. González, A. Martín and E. geometry, submanifolds theory, Chen 5), FIN-00014 University of Helsinki, Sanmartín (Vigo) invariants, harmonic maps, spectral Finland Languages: English, Spanish and French; geometry, Finsler-Lagrange-Hamilton e-mail: [email protected] English is recommended in abstracts and geometry, geometry of PDEs, critical URL: http://www.math.helsinki.fi/~analysis/ in the written version of the communica- point theory and its applications, convexi- NevanlinnaColloquium/ tions ty and optimisation on Riemannian mani- Registration: possible from 1 February to folds, electromagnetic dynamical systems, 21-25: IMACS 2000, Lausanne, 30 April numerical integrator of dynamical systems Switzerland Deadline: for abstracts, 30 November Programme: 30-minute lectures and 15- [International Association for Note: first of a series of conferences minute papers; a workshop engaging Mathematics and Computers World devoted to all aspects of lie theory to be Masters and Ph.D. students in geometry, Congress] held in different locations biannually and an open forum of the Balkan Society Information: contact Prof. Robert Owens, Information: contact I Colloquium on Lie of Geometers IMACS Congress 2000, DGM-IMHEF- Theory and Applications, E.T.S.I. Conference chairs: R. Miron (Romania), LMF, Swiss Federal Institute of Telecomunicación, Universidad de Vigo, Gr. Tsagas (Greece), C. Udriste Technology, CH-1015 Lausanne, 36280 Vigo, Spain; (Romania) Switzerland tel: +86-81-21-52 // +86-81-24-45; fax: Programme committee: M. Anastasiei tel: +41-21-693-35-89; fax: +41-21-693- +86-81-21-16 // +86-81-24-01 (Romania), D. Andrica (Romania), P.L. 36-46 e-mail: [email protected] Antonelli (Canada), Gh. Atanasiu e-mail: [email protected] URL: http://www.dma.uvigo.es/~clieta/ (Romania), D. Blair (USA), N. Blazic URL: http://imacs2000.epfl.ch [For details, see EMS Newsletter 32] 17-22: International Congress of (Yugoslavia), V. Boskoff (Romania), K. Mathematical Physics, London, UK Buchner (Germany), B.Y. Chen (USA), V. 30-2 September: Innovations in Higher Information: Cruceanu (Romania), D. Hrimiuc Education 2000, Helsinki, Finland URL: http://icmp2000.ma.ic.ac.uk/ (Canada), S. Ianus (Romania), L. Theme: higher education in general (not Nicolescu (Romania), D. Opris (Romania), just mathematics) 23-31: ASL European Summer Meeting D. Papuc (Romania), Gh. Pitis (Romania), Information: (Logic Colloquium 2000), Paris, France P. Popescu (Romania), M. Puta e-mail: [email protected] Main themes: proof theory and logical (Romania), H. Shimada (Japan), P. URL: http://www.helsinki.fi/inno2000 EMS September 1999 25 RECENT BOOKS ‘classical’ and ‘modern’ algebraic geome- try. Where they have not enough space, they sometimes omit proofs, but they always RecentRecent booksbooks properly explain all notions and assertions edited by Ivan Netuka and Vladimír Souèek and give references where the correspond- ing proofs can be found. Moreover, at the end of each chapter there are bibliograph- Books submitted for review should be sent to the ous wavelet transform is treated in Chapter ical notes where we find hints for further following address: Ivan Netuka, MÚUK, 3, while Chapter 4, Frames, decribes a gen- reading. The bibliography is large and Sokolovská 83, 186 75 Praha 8, Czech eral framework that allows one to handle includes 350 items. (jiva) Republic. the continuous and the discrete wavelet transforms in a uniform way. Multi-resolu- B. Bollobás, Linear Analysis: An S. K. Berberian, Fundamentals of Analysis, tion analysis with its fast algorithms is pre- Introductory Course, Cambridge Universitext, Springer, New York, 1999, 479 sented in Chapter 5. The construction of Mathematical Textbooks, Cambridge University pp., DM99, ISBN 0-387-98480-1 orthonormal wavelets with compact sup- Press, Cambridge, 1999, 240 pp., £16.95, This book is essentially a record of a course port is given in Chapter 6, and the book ISBN 0-521-65577-3 on functions of a real variable for first-year ends with a brief treatment of spline This is a well-written concise introduction graduate students in mathematics, offered wavelets in Section 6.4. to functional analysis, intended for at the University of Texas at Austin. Numerous illustrations and fully advanced undergraduate students. The The foundational material includes con- worked-out examples further enhance the contents include a nice exposition of stan- struction of the reals, cardinal and ordinal value of this exemplary introduction to the dard material (normed spaces and bound- numbers, Zorn’s lemma and transfinite field. (kn) ed linear operators, Hahn-Banach, induction. Chapter 2 is devoted to Banach-Steinhaus, closed-graph theorem, Lebesgue measure on R introduced via J. Bochnak, M. Coste and M.-F. Roy, Real Stone-Weierstrass, contraction-mapping Carathéodory outer measure, and Chapter Algebraic Geometry, Series of Modern Surveys theorem, Hilbert spaces, orthonormal sys- 3 presents a short introduction to metric in Mathematics 36, Springer, Berlin, 1998, tems, adjoint operators, compact opera- and topological spaces. Chapter 4 deals 430 pp., ISBN 3-540-64663-9 tors, compact normal operators, weak with abstract Lebesgue integral, including This is a substantially enlarged and updat- topologies and duality), as well as unusual convergence theorems, finite signed mea- ed edition of Géométrie Algébrique Réelle by topics like invariant subspaces, fixed-point sures and the Radon-Nikodym theorem. the same authors published in the same theorems (Brouwer, Schauder), the Chapter 5, Differentiation, includes publishing house and in the same series as Bishop-Phelps theorem and geometry of absolutely continuous functions, functions Vol. 12. It is a highly competent mono- finite dimensional spaces. This text, of bounded variation, F. Riesz’s Rising Sun graph written by the leading specialists in although concentrating on abstract spaces, Lemma, indefinite integrals, Lebesgue’s the field. Without any doubt this is an shows the relevance of functional analysis Fundamental theorem of calculus, indispensable book for mathematicians to other areas of mathematics. A nice fea- Lebesgue decomposition of a function of working in real algebraic geometry, but it ture of the book is a large collection of bounded variation and a criterion for is so well written that it can be used as a exercises and notes following each chapter Riemann-integrability. The Stone- textbook for postgraduate students. Of where the most important references are Weierstrass approximation theorem, Lp- course, a preliminary knowledge of com- provided and historical comments present- spaces and real and complex measures are plex algebraic geometry is very helpful for ed. The book will surely be appreciated by studied in Chapter 6. Product measures understanding the differences between ‘a students as well as teachers of mathemati- and Fubini-Tonelli theorem are covered in little classical’ complex algebraic geometry cal analysis. (in) Chapter 7. Chapter 8 deals with Picard’s and real algebraic geometry. But the and Peano’s existence theorems for y’ = development of mathematics is very quick, J. Borwein, P. Borwein, L. Jörgenson and f(x, y). Additional topics in measure and and I can well imagine students who start R. Corless, eds., Organic Mathematics, integration (Jordan-Hahn decomposition to study algebraic geometry via the real CMS Conference Proceedings 20, American of a signed measure, the Radon-Nikodym algebraic geometry. On the other hand, Mathematical Society, Providence, 1997, 412 theorem for σ-finite measures, Lebesgue mathematicians who are at least a little pp., ISBN 0-8218-0668-8 decomposition of measures, convolution) familiar with the complex algebraic geom- This book is the hardcopy version of the are included in Chapter 9. etry will appreciate the new real ideas and electronic ‘Proceedings of the Organic The material is well chosen, the presen- methods. Some chapters of the book, like Mathematics Workshop’ held at Simon tation nice, the balance between special the Nash Functions, will be interesting for Fraser University. The least common and general is reasonable, and each section mathematicians working in analysis, other denominator of the conference was the is accompanied by exercises. This text- parts like Topology of real algebraic varieties, mutual affects of mathematics and modern book will surely find many readers among Algebraic vector bundles, Polynomial or regular technology. We are seeing a steady growth both students and teachers interested in mappings with values in spheres will be inter- in the use of new technology, not only in mathematical analysis. (in) esting for topologists, the chapter Algebraic computation but also in research, teaching ∞ models of C manifolds will attract the atten- and communication (including publica- C. Blatter, Wavelets. A Primer, A. K. Peters tion of differential geometers, while alge- tion). Last but not least, a large portion of Ltd., Natick, 1998, 202 pp., £24, ISBN 1- braists will find here a plenty of interesting the current mathematical knowledge is 56881-095-4 material. encoded in computer algebra systems and This excellent book is intended as an intro- The prerequisities for this book are packages for scientific computation. The duction to the wavelet transform for stu- quite modest. Moreover, the authors have organisers of the conference were interest- dents in mathematics. It provides a solid, included a chapter on ordered fields and ed in the benefits of all these facets of this yet accessible, mathematical foundation real closed fields, where a lot of preparato- modern development, and ‘want the infor- for those interested in learning about ry material is gathered. (We remark that mation of the Proceedings to form exam- wavelets and pursuing the broad range of the book contains much material that is ples of ‘living documents’, connected to applications for which the wavelet trans- more general than the title of the book their references, connected to each other, form has proved successful. suggests; in particular, we find here alge- connected to algorithms for live mathe- The book is divided into six chapters. braic geometry over an arbitrary real matical work on the part of the reader’. The introductory Chapter 1 presents a tour closed field.) It is not surprising that the They want them to be ‘organic’. d’horizon over various ways of signal repre- authors were not able to cover the whole The proceedings contain fifteen invited sentation. Chapter 2 serves primarily as a area of real algebraic geometry. papers and two associated articles. Many tutorial on Fourier analysis. The continu- Nevertheless they cover a lot both from of the underlying papers appeared else- 26 EMS September 1999 RECENT BOOKS where, and each paper has been selected G. Buttazzo, M. Giaquinta and S. The third chapter connects permuta- by author reputation and with the aim ‘to Hildebrandt, One-dimensional Variational tion groups to combinatorial regular struc- select papers with a good potential for Problems, Oxford Lecture Series in tures (such as association schemes, coher- ‘activation’’. Since not every activity dur- Mathematics and its Applications 15, ent configurations, strongly regular and ing the conference can be archived in text Clarendon Press, Oxford, 1998, 262 pp., £35, distance-transitive graphs). form, the editors have attempted at least to ISBN 0-19-850465-9 The fifth chapter is somewhat different personalise the Proceedings by adding a The authors, all of them well-known spe- from the first four, since it concerns infi- short biography of each speaker, with indi- cialists in the field, demonstrate the ideas nite permutation groups, and in particular vidual pictures. (spor) of modern variational calculus in the one- the so-called oligomorphic groups, i.e., dimensional case. It enables them to avoid such permutation groups on an infinite set J.-P. Bourguignon, P. de Bartolomeis and the difficulties connected with multiple Ω for which the number fn(G) of orbits aris- M. Giaquinta (eds.), Geometric Theory of variational integrals and to explain main ing from G when it acts on n-element sub- Singular Phenomena in Partial ideas of the calculus of variations (empha- sets of Ω is finite for all n. Connections with Differential Equations, Instituto Nazionale di sising direct methods) in extremely clear random infinite graphs, with the theory of Alta Matematica Francesco Severi. manner. This project has been realised models (automorphism groups of count- Symp.Mathematica vol. XXXVIII, Cambridge very successfully. The book represents an ably categorical structures) and with grad- University Press, Cambridge, 1998, 182 pp., excellent tool for those who wish to spe- ed algebras are presented with varying £40, ISBN 0-521-63246-3 cialise in the calculus of variations and to level of detail. Attention is also paid to the In May 1995 a workshop on the geometric go on to study more complicated theories growth rate of fn(G) and to preservations of theory of singular phenomena of PDEs of variational problems for multiple inte- linear and circular orders. occuring in real and complex differential grals. At the same time, it is an interesting Students should find this book very geometry was held in Cortona. The book textbook for analysts working in different stimulating because of the many different containing contributions of participants branches who wish to complete their edu- connections it mentions, and this is partic- and related articles is dedicated to Franco cation. The text can be inspiring for grad- ularly true about the fifth chapter. The Tricerri who was a member of the organis- uate students beginning their scientific combinatorial interests of the author obvi- ing committee of the workshop before his activities. I much enjoyed the historical ously influenced the choice of topics and tragic death in a plane crash in China. notes and the vivid style of exposition. one should not expect an introduction to The longest contribution (70 pages) is In Chapter 1, the classical ‘indirect all aspects of permutation groups. an introduction to twistor techniques by P. methods’ based on necessary and sufficient Another feature of the book is its inclu- de Bartolomeis and A. Nannicini. This conditions for optimality are treated. sion of computational group theory. The paper contains a very nice systematic and Chapter 2 gives a framework of function Schreier-Sims algorithm and Jerrum’s fil- detailed description of geometrical prop- spaces indispensable for applying the ter are included in the first chapter and erties of twistor spaces for even-dimension- direct methods (absolutely continuous there is even a short introduction to GAP. al compact connected manifolds with a functions, BV-functions and Sobolev The book contains information about given conformal structure. The paper spaces). Lower semicontinuity methods are resources available on the World Wide ends with a characterisation of manifolds discussed in Chapter 3. In Chapter 4, reg- Web. (ad) for which the corresponding twistor space ularity of minimizers is treated; here also is Kählerian. Glueing procedures for con- the Lavrentiev phenomenon is studied in C. M. Campbell, E. F. Robertson, N. structions of solutions of various geometri- detail. Chapter 5 is dedicated to applica- Ruskuc and G. C. Smith, eds., Groups St cal problems (complete immersed minimal tions, such as the Sturm-Liouville eigenval- Andrews 1997 in Bath, I, II, London surfaces in 3-dimensional Euclidean space, ue problem, the vibrating string, variation- Mathematical Society, Lecture Note Series complete embedded surfaces of constant al problems with obstacles, periodic solu- 260/261, Cambridge University Press, mean curvature and complete conformal tions of variational problems, periodic Cambridge, 1999, 737 pp., £29.95/£29.95, metrics of constant positive scalar curva- soloutions of Hamiltonian systems, non- ISBN 0-521-65588-9/0-521-65576-5 ture on subsets of compact Riemannian coercive variational problems, an existence Two volumes of conference proceedings manifolds) and a discussion of moduli problem in optimal-control theory and contain 64 papers that cover many aspects spaces of such solutions are discussed in a parametric variational problems. In of group theory. The main lectures are paper by R. Mazzeo and D. Pollack. Chapter 6 (Scholia) some ramifications of reflected in the proceedings by five survey The conribution by L. Habermann and calculus of variation and various connec- papers, each 20-50 pages long, concentrat- J. Jost contains a discussion of properties tions to pertinent problems for multiple ing on probabilistic algorithms and geo- of the metric induced on the Teichmüller variational integrals are pointed out. (oj) metric group theory. Similar topics space of (marked) Riemann surfaces Σ of a appear also in many of other longer papers given genus by a choice of a Hermitian P. J. Cameron, Permutation Groups, (those with around 20 pages). metric gΣ on Σ. The paper by M. London Mathematical Society Student Texts 45, The common theme of the probabilistic Giaquinta, G. Modica and J. Souèek Cambridge University Press, Cambridge, 1999, papers is the exploitation of Aschbacher’s describes weak solutions of the variational 220 pp., £15.95, ISBN 0-521-65302-9, classification of maximal subgroups of the problem for the Dirichlet integral on the ISBN 0-521-65378-9 finite classical groups for the purposes of space of maps from a bounded domain in This book is intended as a course that probabilistic recognition in these groups. R³ into the sphere S² with prescribed requires only rudimentary knowledge of Babai and Beals present the black-box boundary values. Two different approach- group theory. It has seven chapters, the group concept and give a number of theo- es are discussed, one in the setting of last comprising tables of simple groups, rems concerning constructions, recogni- Sobolev maps and the other in the setting affine 2-transitive groups and almost sim- tion and description in Monte-Carlo poly- of Cartesian currents. The orbifold funda- ple 2-transitive groups. The sixth chapter nomial time. The paper by Shalev starts by mental group of the Persson-Noether- addresses several stand-alone topics (such citing some classical probabilistic results on Horikawa surfaces is computed in the con- as Blichtfeld’s or Jordan’s theorem) and so symmetric groups and the author com- tirbution by F. Catanese and S. the bulk of the content rests with the first ments on the proof of Dixon’s conjecture Manfredini. five chapters. The first two are standard for simple finite groups. He also presents The book also contains two short papers (regular primitive and multiply transitive some applications to free groups and the by F. Labourie (on solutions of the Monge- groups, wreath products, orbitals, cen- modular group, and points out connec- Ampère equation and its relations to pseu- traliser algebra, and characters) and the tions with profinite groups and Kac-Moody do-holomorphic curves) and by A. M. fourth is dedicated to the O’Nan-Scott algebras, finishing with a sketch of proof of Nadel (existence problem for Kähler- Theorem, with a short sketch of the proof Cameron’s conjecture on the base size of Einstein metrics on a given Fano mani- and with applications (orders and degrees almost simple primitive permutation fold). (vs) of primitive groups, distance-transitive groups. The essay of Praeger is about graphs and some others). primitive prime divisor elements that play EMS September 1999 27 RECENT BOOKS an essential role for recognising algo- £35, ISBN 0-521-64325-2 ument on this way of developing integra- rithms in GL(d, q). This book serves as an introduction to a tion theory. (jama) The longest article is an introductory systematic study of the topic and consists of survey on non-positive curvature in group nine articles by different authors. The best D. Cox, J. Little and D. O’Shea, Using theory by Bridson. He carefully explains description of the content of the book is Algebraic Geometry, Graduate Texts in the notion of the non-positive curvature given by listing the articles and their Mathematics 185, Springer, New York, 1998, and its connection to hyperbolicity, and authors: Introduction to algebraic groups and 499 pp., DM78, ISBN 0-387-98487-9 and 0- develops then the theory of CAT(0) spaces. Lie algebras (R. W. Carter), Weyl groups, 387-98492-5 First he describes properties of groups that affine Weyl groups and reflection groups (R. Many applications of algebraic geometry act properly and cocompactly by isome- Rouquier), Introduction to abelian and methods are demonstrated in this book, tries on CAT(0) spaces, and then gives var- derived categories (B. Keller), Finite groups of which covers a variety of topics related to ious examples of an interaction between Lie type (M. Geck), Generalized Harish- the algorithmic theory of polynomials. In interesting groups and these spaces. Chandra theory (M. Broué and G. Malle), the first chapter basic results and notions The lecture of Brookes blends filtra- Introduction to quantum groups (J. C. about basic algebraic structures, polynomi- tions of group rings with non-commutative Jantzen), Introduction to the subgroup struc- als, Gröbner basis algorithms and affine toric geometry. The developed techniques ture of algebraic groups (M. W. Liebeck), varieties can be found. The second and aim at groups with no non-Abelian free Introduction to intersection cohomology (J. third chapters discuss several approaches subgroups. Nilpotent groups receive spe- Rickard), An introduction to the Lusztig to solving polynomial equations, such as cial attention. Conjecture (S. Donkin). (lbi) elimination theory or resultants. Chapters Among other papers let us mention the 4, 5 and 6 are devoted to topics of classical following three: Bogomolov and Katzarkov C. Constatinescu, W. Filter and K. commutative algebra; the reader can find present a base change construction which Weber, Advanced Integration Theory, here basic facts about local rings, Milnor allows to prove that fundamental groups of Mathematics and its Applications 454, Kluwer numbers, syzygies, Hilbert functions, etc. projective surfaces are rather densely dis- Academic Publishers, Dordrecht, 1998, 861 Chapter 7 deals with geometry polytopes, tributed among all finitely presented pp., £237, ISBN 0-7923-5234-3 toric varieties, Minkowski sums and groups. Ol’shanskii shows that under This volume is devoted to a detailed and Bernstein’s theorem, thereby covering rather weak assumptions, one can find 2- systematic development of the abstract the- some connections between polynomials generated 2-groups (p-groups in some ory of integration. The aim is to build a and convex polytopes. Chapter 8 illus- cases) with a prescribed growth function. logically consistent and advantageous con- trates some applications of Gröbner bases McCammond introduces a generalisation struction, suitable for integration in gener- to problems in integer programming, of small cancellation theory which he has al topological spaces (without local com- combinatorial enumeration problems, applied to the word problem of free pactness or countability assumptions). The spline functions. The last chapter discuss- Burnside groups. (ad) cost is that the terminology in the book dif- es applications from computational alge- fers from the most widely accepted one. bra and algebraic geometry to problems M. Capiñski and E. Kopp, Measure, Among the major features of the authors’ from coding theory. Integral and Probability, Springer approach, let us mention the following: the Reading the book does not require Undergraduate Mathematics Series, Springer, integral is defined via the Daniell construc- more than standard undergraduate knowl- London, 1999, 227 pp., DM59, ISBN 3-540- tion as a functional on a Riesz lattice; the edge. The book can be used in a variety of 76260-4 class of integrable functions is wide due to courses involving solving equations, com- This book aims to present Lebesgue inte- a wide family of negligible sets: null sets for mutative algebra, and their various appli- gral in a way accessible to undergraduate integration are those which are ‘locally cations. It contains a lot of exercises with students as a background to probability insignificant’; measures are finite-valued indications for further development, and theory. This is considered as the main and defined on δ-rings rather than on σ- includes pointers to the abilities of various area of application for the theory and the rings. computer algebra packages. It would be choice of material for the book is influ- The volume is divided into chapters on useful (but not necessary) to read this book enced by this aim. the following topics: vector lattices, defini- in conjunction with Ideals, Varieties and A probabilistic interpretation of dis- tion of the integral, Lp-spaces, real mea- Algorithms by the same authors. (spr) cussed notions is introduced at the begin- sures, the Radon-Nikodym theorem and ning, and leads naturally to the choice of a duality, integration of real functions. The L. Debnath, Nonlinear Partial Differential measure-theoretical approach to the n-dimensional integration is not consid- Equations for Scientists and Engineers, Lebesgue integral. Among the topics cov- ered separately (of course, it falls within Birkhäuser, Boston, 1998, 593 pp., DM138, ered, we mention the law of the large num- the general theory). ISBN 0-8176-3902-0 and 3-7643-3902-0 bers and the central limit theorem. The The text is organised in a definition- One of the major goals of the book is to course is kept at a level accessible to theorem-proof format and each chapter is provide an accessible working knowledge American students at the third year of their endowed with exercises, some of them of some of the current analytical methods studies. Certain notions from functional being important theoretical excursions. required in modern mathematics, physics analysis are also included. The book con- To end with, a brief historical chapter is and engineering. From the immense tains exercises with solutions. The final included. wealth of the world of non-linear PDEs, the Chapter explains the axiom of choice as a There is another book that uses this volume emphasises the part dedicated to tool used to prove the existence of non- approach to integration theory: Integration non-linear wave propagation problems. It measurable sets. Theory: Measure and Integral, by C. contains many new examples of applica- Readers of the book, especially begin- Constantinescu and K. Weber (in collabo- tions in fluid dynamics, plasma physics, ners, will appreciate another strategy of ration with A. Sontag), John Wiley & Sons, non-linear optics, gas dynamics, analytical the authors: instead of the most elegant or 1985, where the old and new definitions dynamics and acoustics. With twenty-one the shortest approach, they prefer what are compared and the advantage of the pages of bibliography, the work can also they consider to be more direct or natural. new approach is explained. The present serve as a reference book for those with a Some assertions are first formulated and book contains more material, but for moti- more detailed interest in some of the con- left to be proved by readers using the hints vation and intuitive remarks the reader is sidered subjects. provided; their proofs are presented later referred to the older book. Chapter 1 is an introduction to linear on. (jiva) The book presents a possibility how to PDEs (method of characteristics, Fourier teach measure and integration theory for method, the use of integral transforms and R. W. Carter and M. Geck (eds.), students whose interest consists in abstract the method of Green’s functions). Chapter Representations of Reductive Groups, mathematics. It also provides also a theo- 2 is dedicated (among other things) to vari- Publications of the Newton Institute, Cambridge retical background for various directions ational principles and the Euler-Lagrange University Press, Cambridge, 1998, 191 pp., in abstract analysis. It is an important doc- equations. Chapters 3-6 are devoted to 28 EMS September 1999 RECENT BOOKS non-linear first-order PDEs. Several sec- in July 1996: Infinite free resolutions (L. L. complex variables, and for mathematicians tions of Chapter 6 discuss properties of Avramov), Generic initial ideals (M. L. who want to familiarise themselves with the solutions of real-world non-linear models, Green), Tight closure, parameter ideals, dual profound ideas of intersection theory. It is including traffic flow, flood waves, chro- geometry (C. Huneke), On the use of local a modern treatise that includes many new matographic models, etc. cohomology in algebra and geometry (P. results, as well as simpler proofs of some In Chapter 7, various aspects of Schenzel), Problems and results on Hilbert older results, and also takes into account Whitham’s equation (non-linear dispersive functions on graded algebra (W. V. the historical development of the subject. waves) are studied. Non-linear diffusion- Vasconcelos), Cohomological degree of graded The text is designed so as to be accessible reaction phenomena are considered in modules (W. V.Vasconcelos). Each series is to anybody equipped with a first course of Chapter 8. Chapter 9 develops the theory close to a monograph and is a good intro- algebraic geometry, assuming that he/she of solitons and the inverse scattering trans- duction to its subject. (tk) consults the appendices at the end of the form; many recent results on the basic book. Moreover, the organisation of the properties of the KdV and Boussinesq G. Friedlander and M. Joshi, Introduction book enables one, after the first six chap- equations are discussed. The non-linear to the Theory of Distributions, 2nd edition, ters, to read the other chapters separately. Schrödinger equation and solitary waves Cambridge University Press, Cambridge, 1998, An important role is played by the are the main focus of Chapter 10. Chapter 175 pp., £42.50, ISBN 0-521-64015-6 and Examples, which appear in large quantities 11 is concerned with the theory of non-lin- 0-521-64971-4 at the end of each section. These illustrate ear Klein-Gordon and sine-Gordon equa- In this book the basic notions of the theory the general theorems and build bridges tions with applications. Non-linear evolu- of Schwartz distributions are explained. It between classical and modern approaches, tion equations and asymptotic methods are is especially convenient for those readers present generalisations and counter-exam- treated in the last chapter. who wish to learn the theory of distribu- ples, and serve to motivate later results. At In a way, the author’s style of exposition tions without a knowledge of functional the end of each chapter, the Notes and follows the idea of R. P. Feynman, quoted analysis, namely of the theory of topologi- References contain historical remarks and in the preface: ‘However, the emphasis cal (locally convex) linear spaces, which is put the material of the chapter into a more should be somewhat more on how to do included as an appendix. The book does general framework. Simply, it is a very the mathematics quickly and easily, and not even require a knowledge of the theo- good book. (jiva) what formulas are true, rather than the ry of the Lebesgue integral. This approach mathematicians’ interest in methods of rig- shows that some very deep theorems can L. Gårding, Mathematics and orous proof.’ A great advantage of this be proved in an elementary way; for exam- Mathematicians. Mathematics in Sweden book lies in the striking balance main- ple, the Schwartz kernel theorem is proved before 1950, History of Mathematics 13, tained between the mathematical and without using the theory of nuclear spaces, American Mathematical Society, Providence, physical aspects of the subject. (oj) referring to the theory of Fréchet spaces 1998, 288 pp., ISBN 0-8218-0612-2 only. There are a lot of exercises supple- This book is an extremely readable history R. W. Easton, Geometric Methods for menting the theory. They contain applica- of mathematics in Sweden up to 1950, writ- Discrete Dynamical Systems, Oxford tions mostly to those ordinary or partial ten by a leading personality from Swedish Engineering Science Series 50, Oxford differential equations to which this theory mathematics. This makes the book differ- University Press, New York, 1998, 157 pp., is applicable, linear differential equations ent from usual books on the history of £50, ISBN 0-195-08545-0 with C∞ coefficients. The basic notions of mathematics: the text brings an expert This small book is an essay on discrete the theory of distributions are not gener- overview of the development of mathemat- dynamical systems and it is very pleasant to alised except for a few pages devoted to ics, presented in a master style that reflects read it. There are no long computations Sobolev spaces. No more general multipli- a deep insight into the subject. Thus the or technically difficult considerations, and cation is introduced than that of a distrib- readers learn important facts from the his- the sampling ideas clearly demonstrate the ution with a C∞ function; for example, the tory of mathematics and extend their pre- view of topological dynamics which was multiplication of a distribution of order n vious knowledge in fields that do not exact- close to C. Conley. Several open problems with a Cn function is not introduced. Only ly overlap their own specialities. are also formulated. The main subject of the last chapter of the book is an excep- It is natural that the history of mathe- the book is the long-time behaviour of tion: it contains harder notions that cannot matics in Sweden is more-or-less a story of orbits. be found in other textbooks on distribu- university towns and their mathematicians. After illustrative examples and basic tions. In this chapter, multiplication of Thus the main body of the book includes a definitions, chain-recurrent sets and the distributions is generalised using the wave- detailed description of mathematics as well Conley decomposition theorem are pre- front sets defined by the growth of their as university life in Uppsala, Lund and sented in Chapter 2. Chapter 3 is devoted Fourier transforms in a given direction. Stockholm. The 18th century and up to to local analysis near hyperbolic invariant Roughly speaking, the product of two dis- 1850 are briefly presented, and the time in sets (stable manifolds, Hartman-Grobman tributions is defined if, in each direction, Uppsala and Lund from 1860 to 1880 is theorem, Smale’s horseshoe and the recent one distribution is C∞. (jjel) described. There is a chapter devoted to notion of resonance zone). The heart of algebraic geometry in Lund before 1900, the book is in Chapters 4 and 5, where iso- W. Fulton, Intersection Theory, 2nd another chapter deals with the situation in lated invariant sets and the Conley index Edition, Ergebnisse der Mathematik und ihrer Uppsala during the period 1860 to 1900 of an isolated block are explained. A dis- Grenzgebiete, Modern Surveys in Mathematics, and whole chapters are devoted to crete analogue of Hamiltonian systems, Springer-Verlag, Berlin, 1998, 470 pp., Bäcklund, Mittag-Leffler, and Mittag- symplectic maps, is studied in Chapter 6 DM178, ISBN 3-540-62046-X Leffler’s and Sonya Kovalevski’s mathe- and the existence of the Birkhoff normal This is the second and unchanged edition matical works. A further chapter describes form is proved. The book ends with the of a famous and fundamental monograph, the development of astronomy and optics notion of an invariant measure and the the first edition of which appeared in in the 19th century. More than a half of the Poincaré recurrence theorem. (jmil) 1984. The author has added some refer- book deals with the mathematics devel- ences that appeared shortly after the first oped in Stockholm (1880-1920 and 1925- J. Elias, J. M. Giral, R. M. Miró-Roig and edition, and refers the reader to the sec- 50), Uppsala (1900-25 and 1930-50) and S. Zarzuela, eds., Six Lectures on ond edition of his book Intersection Theory Lund (1900-50). Here the important con- Commutative Algebra, Progress in in Algebraic Geometry, CBMS 57, Amer. tributions of famous Swedish mathemati- Mathematics 166, Birkhäuser, Basel, 1998, Math. Soc., 1996, for the more recent liter- cians are explained, in particular those of 398 pp., DM128, ISBN 3-7643-5951-X and ature. Bendixon, Phragmén, von Koch, 0-8176-5951-X For young mathematicians interested in Holmgren, Nörlund, Carleman, Pleijel, This volume arises from the following algebraic geometry this book is quite indis- Carlson, M. Riesz, Frostman, Nagell, series of lectures given at the Barcelona pensable. It is also of interest to alge- Beurling and Carleson. This book on Summer School on Commutative Algebra braists, topologists, specialists in several mathematics and mathematicians is EMS September 1999 29 RECENT BOOKS strongly recommended to anybody who All of this offers a chance to enjoy mathe- part is devoted to connections between the likes mathematics and its history. (in) matics as a source of applicable tools that selection, mutation and recombination of are all related one to the other and which genetic information and evolutionary A. J. Hahn, Basic Calculus. From create the beautiful abstract world of ideas, game theory. Special gradients (with Archimedes to Newton to its Role in definitions and theorems. The author has respect to a metric on a simplex), so-called Science, Springer, New York, 1998, 545 pp., written the book both for physics students, Shahshahani gradients, and correspond- DM98, ISBN 0-387-94606-3 to show them the mathematics they use, ing gradient systems are continually used This book is based on calculus courses and for mathematics students, in order to here. delivered at the University of Notre Dame. offer a glimpse of many applications of the A basic knowledge of ordinary differen- As the author remarks, the title of the book abstract ideas they know. The prerequi- tial equations and linear algebra is suffi- could be The story of the calculus. He gives sites comprise just a basic calculus course cient for reading this book. There are five examples of different first-year courses and all advanced undergraduates should many exercises of very different types, that could be based on the book. Other be able to follow the exposition. some providing significant extensions of examples are available on the web site The headings of the eight parts of the the material. The systematic explanation, http://www.nd.edu:80/~hahn/ (the preface book are: finite-dimensional vector spaces exercises and extensive bibliography can contains an obvious URL misprint: a (covering standard linear algebra), infi- stimulate new research. (jmil) hyphen instead of a tilde). This diversity is nite-dimensional vector spaces (including partly due to the fact that a large portion the theory of orthogonal polynomials, B. B. Hubbard, The World According to of college mathematics is also included. Fourier series and Fourier transforms), Wavelets, A. K. Peters, Natick, 1998, 330 pp., The first two chapters are devoted complex analysis (complex series, calculus £28, ISBN 1-56881-072-5 mainly to connections between trigonome- of residues, multivalued functions, and This is one of the best books in the popu- try and astronomy. The third chapter analytic continuation), differential equa- lar mathematical literature, remarkable for describes Archimedes’ area research and tions (mainly second-order equations, its fresh style that makes wavelet history, includes the quadrature of the parabola, including analytical and numerical meth- applications and technical advances inter- while the main subject of the fourth chap- ods), operators on Hilbert spaces (spectral esting for non-mathematicians of any age. ter is analytic geometry; this comprises theory, integral equations and Sturm- The author begins her examination of about one fifth of the book. The next four Liouville systems), Green’s functions (for wavelets with the story of the development chapters deal with the mathematics of ODEs and PDEs), groups and manifolds of Fourier analysis, the essential underpin- Leibniz and Newton. The relation of the (representation theory and tensor analy- ning for telephones, X-ray machines, and rest of material to the history of mathe- sis), Lie groups and their applications (dif- computers. Without any dizzying technical matics is more complicated, but the math- ferential geometry, symmetries of ODEs or mathematical details, she describes the ematics included is easier to describe: it and PDEs, variational calculus and Lie the- more recent meteoric rise of wavelet analy- concerns properties of elementary func- ory). sis and its many practical applications. tions and their use in simple problems This book can be warmly recommended Part II lucidly presents the mathematical involving differentiation and integration. as a basic source for the study of mathe- formulas and details of wavelet analysis for The book contains a lot of material that matics for advanced undergraduates or those seeking a deeper understanding. can be used to relate mathematics to the beginning graduate students in physics In this second edition she includes a dis- evolution of human knowledge. It gives and applied mathematics, and also as a ref- cussion of new medical and genetic appli- various ways, both for students and instruc- erence book for all working mathemati- cations, such as mammography, heart dis- tors, for coping with the elements of calcu- cians and physicists. (jsl) ease and fingerprints. The Part III lus. On the other hand, it is written for Appendices contain proofs of the non-specialists, such as for students of J. Hofbauer and K. Sigmund, Heisenberg uncertainty principle and the informatics, biology, economics, or the Evolutionary Games and Population sampling theorem. Readers who are less humanities. A descriptive style of presenta- Dynamics, Cambridge University Press, sophisticated mathematically will also find tion without formal proofs helps to illus- Cambridge, 1998, 323 pp., £16.95, ISBN 0- a brief review of trigonometry, a list of trate what is going on, but the approach 521-62570-X and 0-521-62365-0 mathematical symbols, and a discussion of eliminates a deeper understanding of the The same authors’ previous book The integrals and the Fourier transform of a nature of mathematics. This is a strong Theory of Evolution and Dynamical Systems periodic function. The bibliography feature of the book as well as its major (Cambridge, 1988) has been used both as a includes technical books and articles. The weakness. The book is more useful for reference book and as a textbook. The author has also included a list of wavelet teaching the history of calculus than for authors have now restructured it and software and electronic recources. teaching the basics of mathematics. Details added a lot of new material to make it clos- This excellent book is a wonderful of the book’s contents can be found on the er to evolutionary game theory. This goal introduction to the world of wavelets. It above URL. (jiva) has forced them to reduce the biological can be strongly recommended to anybody motivation and ideas. The result is a math- interesting in using wavelets. (kn) S. Hassani, Mathematical Physics. A ematical text that divides into four parts. Modern Introduction to its Foundations, The first part, Dynamical systems and O. A. Ivanov, Easy as π?, Springer, New Springer, New York, 1999, 1025 pp., DM179, Lotka-Volterra equations, can be under- York, 1999, 187 pp., DM59, ISBN 0-387- ISBN 0-387-98579-4 stood either as an introduction to popula- 98521-2 This book is a condensed exposition of the tion dynamics and the basic ideas of V. When lecturing to future teachers of math- mathematics that is met in most parts of Volterra or as an introduction to the ematics, possible connections with sec- physics. The presentation attains a very asymptotic behaviour of low-dimensional ondary school mathematics should be good balance between the formal introduc- smooth dynamical systems (mainly of emphasised. Many times the chance is tion of concepts, theorems and proofs on dimension 2 for the continuous case). The lost, because of the lack of good reference one hand, and the applied approach on second part is a concise course in the book where such things are treated in an the other, with many examples, fully or dynamics of non-cooperative games; the accessible way. partially solved problems, and historical properties of replicator equations are used This book is aimed at filling the gap. It remarks. as a main tool. In fact, these equations are offers numerous elementary facts, forming An impressive amount of mathematics equivalent to Volterra-Lotka systems a basis for further development of related is covered. Starting from a set-theoretical (Section 7.5). Global properties of replica- mathematics going far beyond the scope of introduction and linear algebra, through tor equations, mainly permanence and the book. In ten chapters the author deals functional analysis, complex analysis, and persistence, and algebraic properties of with a range of topics, including induction a lot of differential equations (both ODEs interaction matrices (M,B,P-matrices) are and Peano axiomatics, combinatorics and and PDEs, exact and numerical solutions), investigated in the third part and are generating functions, ornaments and geo- up to differential geometry and Lie theory. applied to n-species communities. The last metrical transforms, quaternions, and 30 EMS September 1999 RECENT BOOKS functional equations. Each chapter con- nals. concerned with the classical right quotient tains (mostly solved) problems and related This first of two volumes is self-con- ring U(G) of the von Neumann algebra theorems that extend the reader’s knowl- tained, but assumes graduate real and W(G), as the above conjecture is proved to edge. Exercises help to test the level of complex variable theory with a bit of func- be true in the case when there exists a divi- understanding. The book is based on a tional analysis (as contained in Rudin’s sion ring D between CG and U(G). course taught for several years at St well-known text). In this paperback edi- In one of the longer papers Petersburg University, and will be of inter- tion, a number of small errors and some Mikhajlovskii and Ol’shanskii first give suf- est not only for people oriented towards mathematical mistakes in the original ficient and necessary conditions for an education but as pleasant and interesting hard-cover version have been corrected. HNN-extension of a hyperbolic group G to reading for professional mathematicians. The material is presented with useful pic- be hyperbolic, when A and B are isomor- It can be highly recommended for the tures to help the reader understand the phic infinite elementary subgroups of G mathematical library of any university. subject. (here, ‘elementary’ means cyclic-by-finite). (jiva) Chapter I starts with the Jensen’s for- Using this theorem they show that every mula. Szegõ’s theorem is studied in non-elementary hyperbolic group has a J. Kollár and S. Mori, Birational Geometry Chapter II with the pointwise approximate non-trivial verbally complete quotient that of Algebraic Varieties, Cambridge Tracts in identity property of the Poisson kernel. is a torsion group, and give a similar result Mathematics 134, Cambridge University Press, Chapter III deals with the entire functions with respect to torsion-free hyperbolic Cambridge, 1998, 254 pp., £30, ISBN 0-521- of exponential type (Hadamard factorisa- groups. 63277-3 tion, Lindelöf’s theorem, Phragmén- Further papers include a report by This is a very good introduction to the con- Lindelöf theorems, the Paley-Wiener theo- Carlson on some recent developments in temporary research on algebraic geometry rem, representation of positive harmonic the area of quotient categories of modules called ‘minimal model programme’ or functions as Poisson integrals, Blaschke filtered by complexity, a survey paper by ‘Mori’s programme’. The general idea is products, Levinson’s theorem on the den- Cornick on homological techniques for to find in each birational equivalence class sity of zeros). Quasi-analyticity is studied strongly graded rings, a paper by F. of algebraic varieties a variety that can be in Chapter IV, using Carleman’s criterion Jonson on polysurface groups, a survey by considered as the simplest one. For exam- (including the theorem of Cartan and J. S. Wilson on finitely presented soluble ple, each irreducible curve is birationally Corny). Chapter V contains a discussion of groups, and a paper by R. I. Grigorchuck equivalent to a unique smooth curve. A the moment problem on the real line which starts a systematic investigation of similar investigation of surfaces was started (method based on moment sequences, abstract Tychonoff groups. (ad) at the beginning of this century and result- Carleman’s sufficient condition, a neces- ed in finding (with few exceptions) the sim- sary condition, M. Riesz’s general criterion V. S. Kulikov, Mixed Hodge Structures and plest smooth surfaces in each class, called for indeterminacy). Weighted approxima- Singularities, Cambridge Tracts in ‘minimal models’. It is only in the last two tion on the real line is the topic of Chapter Mathematics 132, Cambridge University Press, decades that efforts have been made to VI (Mergelian’s treatment of weighted Cambridge, 1998, 186 pp., £30, ISBN 0-521- extend this programme to higher dimen- polynomial approximation, Akhiezer’s 62060-0 sions. There one meets many complica- method, Mergelian’s criterion, Pólya’s Let f : (Cn+1, 0) → (C, 0) be a germ of holo- tions; for example, as minimal models one maximum density, the analogue of morphic function with 0 as an isolated sin- must allow not only smooth varieties but Pollard’s theorem, de Branges’s descrip- gularity. The main aim of the book is to also varieties with certain reasonable sin- tion of external unit measures, Krein’s study this singularity by means of ‘global’ gularities. Next, already in dimension 3, functions). A question ‘How small can the methods of algebraic geometry or the the- there have appeared quite new birational Fourier transform of a rapidly decreasing non- ory of analytic spaces. We mention that transformations called ‘flips’ and ‘flops’. zero function be?’ is the title of Chapter VII the ‘global’ appears in this ‘local’ situation So far, many results have been obtained (Levinson’s result, Beurling’s theorems, roughly as follows. Let B be an open ball that are valid in arbitrary dimensions, but Kargaev’s example, Volberg’s work). The in Cn+1 of radius ε, S be an open disc in C the minimal model programme has been final chapter presents the persistence of of radius δ, and S’ = S – {0}. We let X = B completed only in dimension 3, by Mori in the form dx/(1+ x²) with special cases (the ∩ f -1(S) and obtain a map f : X → S. 1988. set of positive lower uniform density, the Finally, setting X’ = X – f -1(0) and restrict- This book gives a very good survey of set of integers, harmonic estimation in slit ing f, we obtain a mapping f’ : X’ → S’. If ε present research, including possible exten- regions). Some discussion of special topics and δ << ε are sufficiently small, then f’ is sions of the minimal model programme. appears in an Addendum. a smooth locally trivial fibration, called a We mention that this programme has also The book is well written and can be rec- Milnor fibration. This is already a very found interesting applications in other ommended to anyone interested in real important ‘global’ object which represents parts of algebraic geometry. The reader is and complex analysis. (pp) the starting point for the investigation of assumed to be familiar with algebraic the singularity; the Gauss-Manin connec- geometry at the level of R. Hartshorne’s O. H. Kropholler, G. A. Niblo and R. tion, mixed Hodge structures, and the the- Algebraic Geometry. The authors have suc- Stöhr (eds.), Geometry and Cohomology in ory of period maps play an important role ceeded in giving a very clear presentation Group Theory, London Mathematical Society here. The book is on the contemporary of the subject. The bibliography has 154 Lecture Note Series 252, Cambridge University research level and will be interesting for items and goes up to 1988. (jiva) Press, Cambridge, 1998, 316 pp., £24.95, specialists in singularity theory, algebraic ISBN 0-521-63556-X and differential geometry. The author P. Koosis, The Logarithmic Integral I, This volume is the proceedings of the assumes the knowledge and training usual Cambridge Studies in Advanced Mathematics Durham Symposium on Geometry and in algebraic and analytic geometry; this 12, Cambridge University Press, Cambridge, Cohomology in Group Theory, held in includes knowledge of sheaf theory and 1998, 606 pp., £29.95, ISBN 0-521-30906- July 1994, and contains 18 articles that the technique of spectral sequences. The 9 and 0-521-59672-6 provide a mixture of new results and sur- book is nicely written, and with the above The theme of this book is the logarithmic veys suggesting a framework for future prerequisites makes this interesting topic integral. The author shows how one can research. The longest survey is by Linnell accessible for postgraduate students. The build up an investigation that explains and who studies an analytical version of the bibliography has 78 items. (jiva) clarifies many different, seemingly unrelat- zero divisor conjecture over C when α (≠ 0) ed problems, from a few simple ideas. is in CG and β (≠ 0) is in Lp(G). He shows T. Y. Lam, Lectures on Modules and Rings, With this book one can begin a serious that for p > 2 one can construct many ele- Graduate Texts in Mathematics 189, Springer, study of real and complex analysis. It can ments α in CG for which there are zero New York, 1999, 557 pp., DM119, ISBN 0- be read as a whole, and presents many divisors β in Lp(G). However, he conjec- 387-98428-3 results, some unpublished, some new, and tures that no such α and β exist when p = This textbook is devoted to the basic parts some available only in inaccessible jour- 2. Much of the discussion of this case is of the modern structural theory of rings EMS September 1999 31 RECENT BOOKS and modules. The first chapter deals with Springer, New York, 1998, 275 pp., DM69, ing are very modest, and the book may free, projective and injective modules and ISBN 0-387-98433-8 and 0-387-98433-X therefore be accessible even to undergrad- the second is devoted to flat modules and The book is designed for those who prefer uate students. A large part of the book is homological dimensions. Chapter 3 treats to ‘go back to the masters’. The organisa- devoted to Examples; these substantially some special questions, such as uniform tion of the book is clear from the titles of extend the theory contained in the basic dimensions, singular submodules and non- its chapters: Geometry: the parallel postu- text, and serve also as exercises and prob- singular rings, as well as the dense sub- late; Set theory: taming the infinite; lems. Because the book naturally deals modules and rational hulls. This is fol- Analysis: calculating areas and volumes; with many formulas, we find at the end a lowed by two chapters on various types of Number theory: Fermat’s last theorem; helpful index of notations (for each chap- rings of quotients. In Chapter 6, Algebra: the search for an elusive formula. ter separately). This book should be avail- Frobenius and quasi-Frobenius rings are Besides the historical development of able in every library. (jiva) investigated, and the last chapter deals these themes, the reader will find the rele- with matrix rings, categories of modules vant parts (in English translation) of the G. Micula and S. Micula, Handbook of and Morita theory, including equivalences works of the most important contributors Splines, Mathematics and Its Applications 462, and dualities. (lbi) to the subject. Kluwer Academic Publishers, Dordrecht, 1999, The book provides exciting reading for 604 pp., £174, ISBN 0-7923-5503-2 E. M. Landis, Second Order Equations of those who wish to see how the subjects This excellent book covers the global theo- Elliptic and Parabolic Type, Translations of evolved in time, what notation or language ry of spline functions and their applica- Mathematical Monographs 171, American was used, and how the various ideas tions to various fields, from the introduc- Mathematical Society, Providence, 1998, 203 unfolded. The book can be recommended tion of the word ‘spline’ by I. J. Schoenberg pp., ISBN 0-8218-0857-5 for anybody interested in the history of the in 1946 to the newest theories of spline- This book is based on the author’s lectures above branches of mathematics. (spor) wavelets and spline-fractals. at Moscow State University. The book is divided into eleven chap- The main tool in Chapter 1 (non-self- J. Lefort, La saga des calendriers ou le fris- ters. Chapter 1 introduces the polynomial adjoint elliptic equations) and in Chapter 3 son millénariste, Pour la Science, Paris, spline functions and their fundamental (parabolic equations) is the use of sub- and 1998, 191 pp., ISBN 2-9029-003-5 properties. Chapters 2 and 3 are devoted super-fundamental solutions, constructed The book deals with the origin and devel- to multivariate and non-linear sets of with the help of Riesz potentials. This opment of different lunar and solar calen- spline functions. The most important approach is used for qualitative studies of dars (Gregorian, Jewish, Chinese, etc.), methods for the numerical solution of inte- the behaviour of solutions near boundary compares them, and presents algorithms gral equations and ordinary differential points (Wiener-type theorems) as well as at for passing from a date in one calendar to equations are treated in Chapters 4 and 5. infinity (Phragmén-Lindelöf and Liouville a date in another, and algorithms for cal- Chapter 6 shows that the most natural theorems). Adding the assumptions of culating the date of Easter. The reader will framework for using spline functions is Cordes type on the spreading of the eigen- find interesting information and get an that of finite element methods. Chapter 7 values of the coefficient matrix, the author astronomical explanation for why it was is a thorough presentation of the finite ele- uses the same method to obtain a priori difficult for our ancestors to measure time. ment method for the solution of boundary estimates for the Hölder norms of solu- The book contains pictures (photographs value problems for partial differential tions. It implies the existence of solutions and reproductions), well-arranged tables equations. Using suitable spaces of spline to boundary value problems for quasi-lin- and mathematical formulas. It is nicely functions, finite element methods for ellip- ear equations. In Chapter 2 (self-adjoint designed and will provide interesting read- tic Dirichlet and Neumann problems and elliptic operators), the estimates for the ing for a wide public. (efas) some non-linear partial differential equa- Hölder norms of solutions are derived with tions are developed. The spline colloca- help of technical tools different from those I. G. MacDonald, Symmetric Functions tion methods for parabolic and hyperbolic used by E. De Giorgi and J. Moser. and Hall Polynomials, 2nd Edition, Oxford problems in two space variables are dis- The Russian edition of the book Mathematical Monographs, Clarendon Press, cussed. Chapter 8 is devoted to spline appeared 26 years ago. Since then, pro- Oxford, 1998, 475 pp., £35, ISBN 0-198- curves, spline surfaces and their B-spline found new results have been obtained, 50450-0 and 0-198-53489-2 representation for computer-aided geo- such as the well-known results of Krylov The first edition of this fundamental metric design; the rational point of view is and Safonov on the estimate of Hölder monograph on symmetric functions and also studied. Chapter 9 briefly describes a norms of the solutions without the assump- Hall polynomials appeared in 1979. A model of shape that combines determinis- tions of Cordes type. However, a signifi- Russian translation was published in 1985. tic splines and stochastic fractals, inherit- cant part of the book presents a non-tradi- This translation was substantially extend- ing their complementary features. The tional approach which is currently of inter- ed, partially in cooperation with the trans- notions of box splines and multivariate est to specialists and cannot be found in lator A. Zelevinsky. The second edition truncated powers are introduced in other monographs and textbooks. appeared in 1995 as a hardback, and rep- Chapter 10. Chapter 11 is a brief intro- There are probably few mathematical resents a further large extension. (The duction to wavelet analysis, and some new texts that present these deep results in Russian translation has 222 pages!) The aspects of the numerical methods using such a vivid and readable manner. Most book under review, published in 1998, is spline wavelets for the solution of evolution sections include bibliographic remarks and an unchanged version of the second edi- partial differential equations are discussed. comments, some of which were added by tion, appearing this time as a paperback. The references section at the end of this the author to the 1997 English edition. I The main aim of the book is to present book aims to be the most exhaustive possi- enjoyed reading them - they reflect briefly the theory of Hall polynomials and their ble (218 pages). All publications known to the exciting way in which the subject has applications. The Hall polynomials are the authors up to August 1998 are listed, been developed. closely connected with symmetric func- and subdivided into three sections: books, The book can be highly recommended tions, and thus the author starts with a monographs and conference reports; orig- to all mathematicians working in the theo- rather long chapter (178 pp.) on symmet- inal papers; and dissertations for a doctor- ry of PDEs of elliptic and parabolic types. ric functions. Indeed, this first chapter al degree or habilitation. The precise and clear exposition also makes the book a standard reference on This book can be strongly recommend- makes it attractive for graduate students. symmetric functions. The Hall polynomi- ed to researchers and graduate students (oj) als were discovered only in the 1950s, but involved in numerical methods and com- have turned out to play an important role putation, approximations, differential R. Laubenbacher and D. Pengelley, in many areas of mathematics. For this equations, and integral equations. (kn) Mathematical Expeditions. Chronicles by reason, MacDonald’s book is interesting the Explorers, Undergraduate Texts in for mathematicians working in various J. C. Migliore, Introduction to Liaison Mathematics. Readings in Mathematics, directions. The prerequisities for the read- Theory and Deficiency Modulus, Progress in 32 EMS September 1999 RECENT BOOKS Mathematics 165, Birkhäuser, Boston, 1998, 850391-1 215 pp., DM98, ISBN 0-8176-4027-4, ISBN This volume presents for the first time 140 J. J. Risler (directeur de la publication), 3-7643-4027-4 letters from Sylvester’s correspondence, Matériaux pour l’histoire des mathéma- This is a highly specialised monograph selected from some 1200 letters from many tiques au XXe siècle, Séminaires & Congrès that provides a very good introduction to archives, libraries and private collections 3, Société Mathématique de France, Paris, contemporary research in the fields of liai- in the USA, Great Britain, Germany and 1998, 282 pp., ISBN 2-85629-065-5 son theory and deficiency modules. It will France. These letters cover Sylvester’s life This book is the third volume of materials be of interest first to algebraic geometers and work and provide a detailed look at his for the history of mathematics in the XXth and algebraists. thoughts and thought processes. They century, dedicated to the memory of Jean Liaison (or linkage) is an equivalence show him in both personal and profession- Dieudonné (1906-92). The volume con- relation among subschemes of given al spheres during his 82 years. The letters tains a short introduction that describes dimension in a projective space Pn over an reflect his research activities, the range of the ‘history’ of the book. In January 1996, algebraically closed field. Two subschemes his correspondents, and his interests. on the occasion of the inauguration of the are directly CI-linked if their union is a The letters are divided into six chap- new pavilion of Laboratoire J. A. complete intersection, and directly G- ters. Each chapter opens with a short pre- Dieudonné at the Université de Nice - linked if their union is arithmetically lude that gives a thematic overview of its Sophia Antipolis, there was a conference Gorenstein. While the first notion is rela- subject. An extensive historical and math- organised by the University, l’Académie de tively classical, the second is rather new ematical commentary accompanies the let- Sciences, d’Institut des Hautes Études and represents an area of current research. ters, and references to pertinent secondary Scientifiques et du Centre National de la The deficiency modules are defined for a literature (mathematical and historical) Recherche Scientifique. The book contains closed subscheme V of Pn of dimension r and bibliography are included. Chapter 1 the conference programme, with eleven i as cohomology modules H* (IV), 1 ≤ i ≤ r, of contains letters from 1834 to 1849, contributions in French and English deliv- the ideal sheaf of V, and measure the fail- Chapter 2 spans the years 1850 to 1854 ered at the conference. They are devoted ure of V to be arithmetically Cohen- when Sylvester formulated his theory of to fundamental ideas of 20th-century Macaulay. They appear as an important invariants, the 22 years from 1855 to 1876 mathematics, such as hyperbolic equations, tool for the study of the liaison. To read are covered by Chapters 3 and 4, Chapter functional analysis, homotopy theory of this book the reader needs an introductory 5 is on Sylvester at Johns Hopkins fibre spaces, group representations, sheaf knowledge of algebraic geometry and com- University (1876 to 1883) and the final theory, the mathematical theory of mutative algebra, but the author tries to chapter is devoted to the years 1884 to Brownian motion, and Hilbert’s twelfth help as much as possible. The first chapter 1897. (mnem) problem. At the end, an alphabetical index ‘Background’ presents some less standard is included. notions and results needed in the book. A. Pasini, Elementi di Algebra e Geometria. This book can be recommended to all The author pays great attention to moti- I. Naziono di base, II. Elementi di Algebra, mathematicians interested in the history of vation and the geometric aspects of the III. Algebra lineare e Geometria, Liguori modern mathematics. (mnem) theory. There are many examples through Editore s.r.l., Napoli, 1998, 179/183/524 pp., which the reader is introduced into the 22000L/22000L/ 49000L, ISBN 88-207- P. C. Roberts, Multiplicities and Chern theory, thereby stimulating research inter- 2739-0, 88-207-2740-4 and 88-207-2741-2 Classes in Local Algebra, Cambridge Tracts est in this field; some of these are comput- This book is designed for students who in Mathematics 133, Cambridge University ed using the program ‘Macaulay’. The have begun to study university mathemat- Press, Cambridge, 1998, 303 pp., £37.50, book will be useful both for specialists and ics; it is divided into three volumes. ISBN 0-521-47316-0 for postgraduate students in these areas. The first volume contains basic notions It is well known that the development of The bibliography includes 203 items and of mathematics (sets, maps, relations, algebraic geometry greatly influenced goes up to 1997. (jiva) logic, finite sets, binomial coefficients, progress in algebra. In recent decades, the mathematical induction, infinite sets, reverse trend appeared and proved to be A. D. Osborne, Complex Variables and countable sets, axiom of choice, cardinal very fruitful; namely, methods of algebraic their Applications, International Mathematics numbers). The second volume covers the geometry started to be applied within the Series, Addison Wesley Longman, Harlow, necessary algebraic background (fields, framework of pure algebra. The main idea 1998, 454 pp., £19.95, ISBN 0-201-34290- rings, groups, semigroups); substantial consists in substituting objects of algebraic 1 parts of this volume are devoted to the real geometry by more general objects in pure The author’s intention was to write a versa- and complex fields, and to polynomials in algebra. This book represents an exposi- tile book suitable for both first and second one and several variables. tion of some parts of this modern and courses in complex analysis both for math- The third volume is devoted to linear rapidly developing branch of algebra. ematicians and for those interested in algebra and geometry. It begins with clas- An important notion in algebraic geom- applications. The only assumed knowl- sical material on vector spaces and linear etry is that of multiplicity of intersection. edge is calculus and basic real analysis. All maps (subspaces, linear independence, The author uses two notions of multiplici- important techniques and applications are bases, dimension, direct sums, etc.) and on ty. The first generalises the multiplicity of covered. The book contains a range of basic applications, such as systems of linear intersection of a plane curve with itself and exercises from standard to challenging equations. Also included are inner prod- is called ‘Samuel multiplicity’. The second questions; there is a manual providing full uct vector spaces (orthogonal bases, generalises the multiplicity of intersection solutions to all questions, available to lec- orthogonalisation, etc.). The next three of two subvarieties of a variety, and its turers from the publishers. In addition to chapters present standard material on requires the use of homological algebra; it classical material, the book includes affine and Euclidean geometry. In the is called ‘Serre multiplicity’. Most of this asymptotic series and elliptic functions as final chapters, linear algebra and geometry book is devoted to the introduction and well as applications to ODEs and integral are presented as closely linked subjects; a investigation of these (and other) multi- transforms. Everything is carefully central idea is the matrix representation of plicities. For these purposes a lot of tech- explained with a great sense for clear and linear maps (algebra of linear maps, alge- niques are developed. We mention in par- straightforward approach, accompanied by bra of matrices, eigenvalues and eigenvec- ticular the Chern classes of locally free numerous short historical remarks. Many tors, orthogonal and unitary transforma- sheaves, Chern character, and local Chern pictures and very good typographical pre- tions and matrices, affine transformations, characters. All these notions were inspired sentation make reading pleasant end transposed and conjugate complex matri- by the corresponding notions of algebraic enjoyable. (jiva) ces, bilinear and quadratic forms, conic topology, but they are defined in purely sections and quadrics). algebraic terms. K. H. Parshall, James Joseph Sylvester: Each volume includes exercises and an The book is nicely written, and serves Life and Work in Letters, Clarendon Press, index. They are well written and can be well as an introduction to this interesting Oxford, 1998, 321 pp., £55, ISBN 0-19- recommended for basic study. (jbe) branch of algebra, as well as containing EMS September 1999 33 RECENT BOOKS very recent results. The necessary prereq- Vol. 2, Cambridge Studies in Advanced tion approach is developed; various norms uisities from algebra and algebraic geome- Mathematics 62, Cambridge University Press, are used as performance measures. try are rather modest, but a good back- Cambridge, 1999, 581 pp., £45, ISBN 0-521- Chapters 4 and 5 are devoted to an opti- ground in algebraic geometry gives the 56069-1 mal design for tracking some signals or reader better motivation. There are many The book is a continuation of Volume 1 rejecting disturbances. It is supposed that examples to make the exposition more (1986), starting with Chapter 5, Trees and the nominal plant G and the nominal con- attractive, and there are exercises at the the composition of generating functions, and troller K are augmented by a plant G and a end of each chapter. The theory present- continuing with Chapter 6, Algebraic, D- controller Q. Optimisation techniques are ed here is a beautiful interplay of several finite, and noncommutative generating func- developed for both types of designs: off- branches of mathematics and provides tions, and Chapter 7, Symetric functions. The line and iterated or recursive (Q, S)- good reading for algebraists, geometers, last chapter has two appendices: ‘Knuth designs (for S-parametrisation). The next and topologists. (jiva) Equivalence, Jeu de taquin, and the two chapters describe direct and indirect Littlewood-Richardson rule’ (by S. Fomin) adaptive-Q controls; in the latter, con- K. A. Ross, J. M. Anderson, G. L. and ‘The characters of GL(n, C)’. Each troller designs are based on on-line identi- Litvinov, A. I. Singh, V. S. Sunder and N. chapter has its own set of exercises with fied models. As the adaptation proceeds J. Wildberger (eds.), Harmonic Analysis solutions, 261 in all but many more when slowly compared to the plant dynamics, and Hypergroups, Trends in Mathematics, sub-exercises are counted. The references averaging techniques are used. In Chapter Birkhäuser, Boston, 1998, 249 pp., DM208, to them and the bibliographies to the 8 it is shown how to apply the direct adap- ISBN 3-764-33943-8, ISBN 0-817-63943-8 chapters constitute a giant survey of litera- tive-Q algorithm to non-linear systems by This book contains the proceedings of a ture on enumerative combinatorics. means of linearisation. The concluding conference on harmonic analysis on hyper- What else is to be added to our com- two chapters are closer to real applications: groups held in December 1995 in Delhi. ments on this excellent book? Perhaps a the real-time implementation is discussed Much of this book is concerned with the quotation from G.-C. Rota’s foreword: and three examples (a disc driver control notion of hypergroups. Let M(H) be the ‘Every once in a long while, a textbook system, control of a heat exchanger and a space of all bounded Borel measures on a worthy of the name comes along; ... flight control system) are studied in detail. locally compact topological space H. If H Weber, Bertini, van der Waerden, Feller, The book is directed to graduate stu- is a group, then there is a canonical prod- Dunford and Schwartz, Ahlfors, Stanley.’ dents in system theory; beginners are first uct * defined on M(H) by δx * δy = δxy. For (mkl) recommended to read the book ‘Feedback a general H, if * is a product on M(H) sat- Control Theory’ by J. Doyle, B. Francis isfying a certain set of axioms (generalising T.-T. Tay, I. Mareels and J. B. Moore, and A. Tannenbaum (Macmillan, New properties of M(H) from the group case), High Performance Control, Systems & York, 1992). The proofs are sometimes the couple (H,*) is called a hypergroup. Control: Foundations & Applications, sketchy and more attention is paid to moti- Many results in classical harmonic analysis Birkhäuser, Basel, 1998, 344 pp., sFr148, vation and to system philosophy. The (the Fourier transform, the Plancherel and ISBN 3-764-34004-5 and 0-817-64004-5 extensive bibliography with comments on inversion formulas, the Plancherel theo- The main theme of this book is to deter- the references is also very useful. This rem) can be extended to hypergroups. mine whether high performance can be book, together with (e.g.) I. Meerels and J. Signed hypergroups and relations to achieved in the face of uncertainity. The Polderman’s ‘Adaptive Control Systems’ Markov chains are discussed in the contri- first three chapters are introductory, with (Birkhäuser, Basel, 1996), provides a good bution by K. A. Ross. Connections among attention paid to the description of all sta- account of the research that has been done characters of hypergroups, families of bilising controllers for which the factorisa- during the last decade. (jmil) orthogonal polynomials (such as Legendre polynomials) and the Sturm-Liouville problem appear in the three lectures by A. L. Schwartz. An extension of the wavelet Centre de Recerca Matematica (Barcelona) transform to hypergroups is discussed in List of visitors, September - December 1999 the paper by K. Trimeche. An approach to non-commutative harmonic analysis on a J. L. Balcazar, Barcelona, 5 Sep - 10 Dec 99, Computational theory Lie group, based on an old idea of K. Baranski, Warszawa, 1 Dec 98 - 31 Oct 99, Analysis Frobenius, and its relation to hypergroups M. Brunella, Dijon, 7 - 30 Nov 99, Geometry is described in the contribution of N. J. A. Candel, California, 22 Nov - 20 Dec 99, Geometry Wildberger. W. Chacholski, Connecticut, 1 - 31 Oct 99, Algebraic topology Other topics include De Branges mod- J. A. Crespo, Barcelona, 1 Sep 99 - 31 Jul 00, Algebraic topology ules (S. Agrawal and D. Singh), moment M. Cruz, Mexico, 10 Nov - 10 Dec 99, Geometry functions on hypergroups (L. Gallardo), F. X. Dehon, Paris, 1 Oct 99 - 30 Sep 01, Algebraic topology behaviour of the Plancherel measure and S. Dumitrescu, Lyon, 1 - 20 Nov 99, Geometry multiplier theorems (M.-O. Gebuhrer), K. Faure, Toulouse, 1 Sep 99 - 30 Jun 00, Geometry disintegration of measures (H. Helson), J. M. Gambaudo, Dijon, 7 - 20 Nov 99, Geometry multipliers of de Branges-Rovnyak spaces F. Gautero, Valbonne, 14 Sep 98 - 30 Sep 00, Dynamical systems (B. A. Lotto and D. Sarason), applications E. Ghys, Lyon, 1 - 21 Nov 99, Geometry to measures on compact spaces (R. Nair), A. Guillot, Lyon, 1 - 20 Nov 99, Geometry applications to functional equations (H. P. Koskela, Jyvaskyla, 18 Jan - 31 Dec 99, Analysis Stetkaer), actions of finite hypergroups on M. Lagrange, Dijon, 8 - 20 Nov 99, Geometry finite sets (V. S. Sunder and N. J. S. Lamy, Prest, 1 Sep 99 - 30 Jun 00, Geometry Wildberger), positivity of Turán determi- J. J. Loeb, Angers, 1 - 15 Nov 99, Geometry nants (R. Szwarc), relations to semigroups F. Loray, Villeneuve d’Ascq, 24 Oct - 20 Nov 99, Geometry of positive definite functions (M. E. Walter) M. McQuillan, Oxford, 7 Nov - 3 Dec 99, Geometry and limit theorems for random walks (H. M. Mimura, Okayama, 1 Oct - 30 Nov 99, Algebraic topology Zeuner). I. Morrison, New York, 1 Jan - 31 Dec 99, Applied mathematics The book contains nice survey papers H. R. Morton, Liverpool, 1 - 31 Oct 99, Applied mathematics on properties of hypergroups, as well as F. Sanchez-Bringas, Mexico, 1 - 15 Dec 99, Geometry papers presenting applications and rela- J. Seade, Mexico, 30 Nov - 14 Dec 99, Geometry tionships with many different parts of C. Tarquini, Rennes, 1 Sep 99 - 30 Jun 00, Geometry mathematics. (vs) A. Verjovsky, Mexico, 10 Nov - 15 Dec 99, Geometry X. Zhang, Beijing, 1 Mar 99 - 28 Feb 00, Dynamical systems R. P. Stanley, Enumerative Combinatorics, 34 EMS September 1999 Journal of the European Mathematical Journal
The first issue of this EMS journal appeared in January 1999. The contents of the first two issues were given in the June Newsletter. Further information can be obtained by e-mail: [email protected]
Volume 1, Number 3 F. Lin and T. Rivière, Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents J. Kollár, Effective Nullstellensatz for arbitrary ideals Erratum to M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups
Call for Monographs Portuguese Mathematical Society (SPM) Editor-in-Chief: A. B. Cruzeiro Editorial board: I. Fonseca, E. Marques de Sà, M. Pollicott, J. Rezende
This new series will publish research monographs of high scientific level in all domains of mathematics. Each mono- graph should be around 100 pages at least and, as far as possible, present a self-contained exposition of the subject. These can be written in English, French or Portuguese, but preferably in English. Proposals should be addressed to A. B. Cruzeiro, Grupo de F’sica Matemàtica da Universidade de Lisboa, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal. tel: +351-1-790-4726; fax: +351-1-795-4288 e-mail: [email protected]
Post-doctoral Positions in Stochastic Analysis We wish to draw the attention of strong young researchers in stochastic analysis to the possibili- ty of research positions funded by the TMR (Training-Mobility-Research) contract in Stochastic Analysis ERB-FMRX-CT96-0075 of the European Union.
There are currently up to four one-year positions available. These positions are to carry out research within one of the teams of the contract, and the researcher would be based at one of the following laboratories: (Prof. T. J. Lyons) Mathematics Dept, Imperial College, Huxley Building, 180 Queens Gate, London SW7 2BZ, UK ([email protected]) (Prof. D. Elworthy) Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK ([email protected]) (Prof. Y. Le-Jan) Mathematiques, Université de Paris-Sud, Centre d’Orsay, 91405 Orsay, Cedex, France ([email protected]) Each institution has a top-quality mathematics department, and within it, has strong research activity in probability and stochastic analysis. Anyone who is well qualified and interested in one of the positions should get in immediate contact with the relevant laboratory mentioned above and also send a note recording their interest to Prof. T.J. Lyons at Imperial College.
Please note that the European Union imposes strong nationality, and age restrictions on these positions. Researchers must be a national of a member State or an associated State of the Community (Austria, Belgium, Denmark, Finland, France, Germany, Greece, Iceland, Ireland, Italy, Liechtenstein, Luxembourg, Netherlands, Norway, Portugal, Spain, Israel, Sweden and the UK) and not from the country where the position is to be held. Details can be found on http://sagwww.ma.ic.ac.uk/.
EMS September 1999 35